Sindbad~EG File Manager

Current Path : /usr/home/beeson/public_html/michaelbeeson/research/papers/
Upload File :
Current File : /usr/home/beeson/public_html/michaelbeeson/research/papers/SevenTriangles.dvi

����;� TeX output 2008.11.27:1057������header=pstricks.pro�header=pst-dots.pro�y�������>�color push gray 0�Y�	color pop���?��>����color push gray 0�4����F�color push gray 0�	color pop���F�src:542 SevenTriangles.tex�D��tG�G�cmr17�No�7ttriangle�can�b�s�e�decomp�osed�in��qto�sev�en����jk�congruen��qt�7ttriangles������������X�Qcmr12�Mic��rhael��Beeson����������{�UNo��rv�em�b�S�er��27,�2008��$�����!K�color push gray 0�	color pop����!K�src:547 SevenTriangles.tex�t�:		cmbx9�Abstract���э���
��color push gray 0��	color pop���&��src:548 SevenTriangles.tex�o���		cmr9�W��:�e�d�in��9v�estigate�d�the�problem�of�cutting�a�triangle�in��9to��5��"		cmmi9�N�]��congruen�t����triangles.��:While�`�this�can�b�A�e�done�`�for�certain�v��|ralues�of��N����,�s�w��9e�pro�v�e�that����it�`�cannot�b�A�e�`�done�for��N���=��7.��TThis�result�is�a�sp�ecial�case�`�of�m��9uc�h�`�more����general��mresults��lobtained�in�[�1����],��but�the�pro�A�of�in�this�pap�A�er�ma��9y�still�b�A�e����of��7some�in��9terest,�߰b�A�ecause�only�metho�ds�of��8Euclidean�geometry�are�used����(including��Osimple�trigonometry�that�can��Nin�principle�b�A�e�done�b��9y�geometric����argumen��9ts).����������N�ffcmbx12�1��1L�In���tro�s3duction��阍��src:556 SevenTriangles.tex�W��:�e�ˈconsider�the�ˉproblem�of�cutting�a�triangle�in��9to��N��l�congruen�t�ˈtriangles.����Figures�Z@1�through�1�sho��9w�that,�k{at�ZAleast�for�certain�triangles,�this�can�b�A�e����done��with���N��<�=��W3,��4,�5,�6,��9,�and�16.�7�Suc��9h��a�conguration�is�called�an�����N����-tiling.��v�E������color push gray 0�i?E��������x*��color push gray 0�K�`y

cmr10�Figure�UU1:�q�Tw���o�3-tilings��	color pop���XM~��src:563 SevenTriangles.tex���s��h" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 170.71652 0.0 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 42.67912 73.92262 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 73.92262 moveto 170.71652 0.0 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 85.35826 49.28131 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  85.35826 49.28131 moveto 85.35826 0.0 L 0 setlinecap stroke  end ������<�g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 85.35826 0.0 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 42.67912 73.92262 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 73.92262 moveto 85.35826 0.0 L 0 setlinecap stroke  end �k" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 42.67912 24.6413 L 0 setlinecap stroke  end �u" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 73.92262 moveto 42.67912 24.6413 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 24.6413 moveto 85.35826 0.0 L 0 setlinecap stroke  end �����	color pop��@��&��src:606 SevenTriangles.tex�The�ʠmetho�A�d�illustrated�ʡfor��N����=���4�,9,�ّand�16�clearly�generalizes�to�an��9y����p�A�erfect�"gsquare�"f�N����.��vWhile�the�exhibited�3-tiling,�R�6-tiling,�and�5-tiling�"gclearly����dep�A�end�-(on�the�exact�-'angles�of�the�triangle,�[���j��		cmti9�any��triangle�can�b�A�e�decomp�osed����in��9to�\��n���-=��Aa�cmr6�2���S�congruen�t�triangles�b�y�dra�wing��n�=������		cmsy9���1�lines,�n�parallel�to�eac��9h�edge����and�*dividing�)�the�other�t��9w�o�*edges�in�to�*�n��equal�parts.�ZsMoreo��9v�er,�/*the�*large����(tiled)��|triangle��}is�similar�to�the�small�triangle�(the�\tile").��It�follo��9ws�that����if�Rw��9e�ha�v�e�a�Rtiling�of�a�triangle��AB�r�C����in�to�R�N�J��congruen�t�triangles,�a@and��m�����is��an��9y�in�teger,��~w�e�can�tile��AB�r�C�b��in�to��N���m���-=�2����triangles�b�y�sub�A�dividing�the�����>�color push gray 0����1��Y�	color pop����*�y�������>�color push gray 0�Y�	color pop���?���C?E��>�����color push gray 0�i?E��������G�d�color push gray 0�Figure�UU2:�q�A�4-tiling,�a�9-tiling,�and�a�16-tiling�	color pop���XM~��src:572 SevenTriangles.tex���9��g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 85.35826 0.0 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 42.67912 73.92262 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 73.92262 moveto 85.35826 0.0 L 0 setlinecap stroke  end �t" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 36.9613 moveto 64.01869 36.9613 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 36.9613 moveto 42.67912 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 0.0 moveto 64.01869 36.9613 L 0 setlinecap stroke  end ������	��g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 85.35826 0.0 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 42.67912 73.92262 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 73.92262 moveto 85.35826 0.0 L 0 setlinecap stroke  end �t" tx@Dict begin STP newpath 0.8 SLW 0  setgray  14.32059 24.6413 moveto 71.13144 24.6413 L 0 setlinecap stroke  end �v" tx@Dict begin STP newpath 0.8 SLW 0  setgray  28.45232 49.28131 moveto 57.85413 49.28131 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  14.32059 24.6413 moveto 28.45232 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  28.45232 49.28131 moveto 57.85413 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  28.45232 0.0 moveto 57.85413 49.28131 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  57.85413 0.0 moveto 71.13144 24.6413 L 0 setlinecap stroke  end �������/�g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 85.35826 0.0 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 42.67912 73.92262 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 73.92262 moveto 85.35826 0.0 L 0 setlinecap stroke  end �v" tx@Dict begin STP newpath 0.8 SLW 0  setgray  10.66977 18.48065 moveto 74.68848 18.48065 L 0 setlinecap stroke  end �t" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 36.9613 moveto 64.01869 36.9613 L 0 setlinecap stroke  end �u" tx@Dict begin STP newpath 0.8 SLW 0  setgray  32.00934 55.44197 moveto 53.3489 55.44197 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  10.66977 18.48065 moveto 21.33955 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 36.9613 moveto 42.67912 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  32.00934 55.44197 moveto 64.01869 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 0.0 moveto 53.3489 55.44197 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 0.0 moveto 64.01869 36.9613 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  64.01869 0.0 moveto 74.68848 18.48065 L 0 setlinecap stroke  end �����	color pop��d,ፑ>����color push gray 0�X,፟��捍�v���color push gray 0Figure�UU3:�q�Three�4-tilings�	color pop���G;��src:583 SevenTriangles.tex���g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 98.56349 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  98.56349 0.0 moveto 98.56349 56.9055 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  49.28174 28.45274 moveto 49.28174 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  49.28174 28.45274 moveto 98.56349 0.0 L 0 setlinecap stroke  end �k" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 98.56349 56.9055 L 0 setlinecap stroke  end �v" tx@Dict begin STP newpath 0.8 SLW 0  setgray  49.28174 28.45274 moveto 98.56349 28.45274 L 0 setlinecap stroke  end �����qϞ�g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 98.56349 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  98.56349 0.0 moveto 98.56349 56.9055 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  49.28174 28.45274 moveto 49.28174 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  49.28174 28.45274 moveto 98.56349 0.0 L 0 setlinecap stroke  end �k" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 98.56349 56.9055 L 0 setlinecap stroke  end �u" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.92262 14.22636 moveto 98.56349 56.9055 L 0 setlinecap stroke  end ������<�g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 98.56349 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  98.56349 0.0 moveto 98.56349 56.9055 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  49.28174 28.45274 moveto 49.28174 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  49.28174 28.45274 moveto 98.56349 0.0 L 0 setlinecap stroke  end �k" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 98.56349 56.9055 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  98.56349 0.0 moveto 73.92262 42.67912 L 0 setlinecap stroke  end �����	color pop����W�rst�e�tiling,�y�replacing�e�eac��9h�of�the��N�^��triangles�b��9y��m���-=�2���E�smaller�ones.�
OHence����Wthe� �set�of��N���for�whic��9h�an��N����-tiling� �of�some�triangle�exists�is�closed�under����Wm��9ultiplication�Tb�y�squares.����d��src:614 SevenTriangles.texLet�4]�N�-@�b�A�e�of�the�4\form��n���-=�2���:�+�͓�m���-=�2��*��.�y�Let�triangle��T�}@�b�A�e�a�righ��9t�triangle����Wwith�'�p�A�erp�endicular�sides��n�'��and��m�,�,<sa��9y�with��n��P���Q�m�.�SiLet��AB�r�D�in�b�A�e�a�righ��9t����Wtriangle��with��base��AD�$�of�length��m���-=�2��*��,�the�righ��9t�angle�at��D�$�and�altitude����W�mn�,�r�so�`side�`�B�r�D����has�length��mn�.���Then��AB�D����can�b�A�e�`decomp�osed�`in��9to��m����W�triangles��&congruen��9t��%to��T�H��,�Zarranged�with�their�short�sides�(of�length��m�)����Wparallel��_to��`the�base��AD�A��.�
tNo��9w,��*extend��AD�!'�to�p�oin��9t��`�C����,��*lo�cated��n���-=�2��
�past��D��.����WT��:�riangle���AD�A�C����can��b�e�tiled�with��n���-=�2��C~�copies�of���T�H��,��arranged�with�their�long����Wsides�V�parallel�to�V�the�base.���The�result�is�a�tiling�of�triangle��AB�r�C����b��9y��n���-=�2���t�+����m���-=�2�����W�copies�L�of�L��T�H��.��vThis�is�a�rigid�tiling.��wThe�5-tiling�exhibited�in�Fig.��v3�is�the����Wsimplest��example,��where��n����=�2��and���m��=�1.��The�case���N����=�13�=�3���-=�2��
d�+�߽2���-=�2��(��is����Willustrated�Tin�Fig.�p4.����d��src:628 SevenTriangles.texIf�%ythe�original�triangle��AB�r�C��a�is�%zc��9hosen�to�b�A�e�isosceles,�i�then�eac�h�of����Wthe���n���-=�2��!��triangles��can�b�A�e�divided�in�half�b��9y�an�altitude;�.hence�an�y�isosceles����Wtriangle�i�can�b�A�e�i�decomp�osed�i�in��9to�2�n���-=�2���;�congruen�t�triangles.�/If�the�original����Wtriangle���is���equilateral,��then�it�can�b�A�e�rst�decomp�A�osed�in��9to��n���-=�2��(u�equilateral����Wtriangles,���and���then�these�triangles�can�b�A�e�decomp�osed���in��9to�3�or�6�triangles����Weac��9h,���sho�wing��that��an�y�equilateral�triangle��can�b�A�e�decomp�osed��in��9to�3�n���-=�2���n�or����W6�n���-=�2�����congruen��9t�})triangles.�S�Note�}*that�these�are�dieren��9t�tilings�than�those����Wobtained�_b��9y�_the�metho�A�d�of�the�rst�paragraph�of�this�section.�߱F��:�or�example����Ww��9e��3can��212-tile�an�equilateral�triangle�in�t��9w�o��3dieren�t�w�a�ys,���starting�with��3a����W3-tiling��and�then�sub�A�dividing�eac��9h��triangle�in�to�4�triangles�(\sub�A�dividing����Wb��9y�T4"),�or�starting�with�a�4-tiling�and�then�sub�A�dividing�b�y�3.����d��src:636 SevenTriangles.texThe���elemen��9tary���constructions�just�describ�A�ed�suce�to�pro�A�duce��N����-����Wtilings�D|when�D{�N�=`�has�one�of�the�forms��n���-=�2��*��,�PF�n���-=�2��XN�+�-��m���-=�2���,�2�n���-=�2���,�PE3�n���-=�2���,�or�D|6�n���-=�2���.���The����Wsmallest�Z�N�=�not�of�Yone�of�these�forms�is��N���=�&�7.���The�main�theorem�of����Wthis��Rpap�A�er��Qis�that�there�is�no�7-tiling.��iIn�[�1����],���w��9e�ha�v�e��Qcompletely�solv�ed�����>�color push gray 0����2��Y�	color pop����f�y�������>�color push gray 0�Y�	color pop���?���9���>����color push gray 0�_�����ˍ������color push gray 0�Figure�UU4:�q�A�5-tiling�	color pop���N5��src:591 SevenTriangles.tex���8���g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 156.4899 0.0 L 0 setlinecap stroke  end �m" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 125.19193 62.59596 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  125.19193 62.59596 moveto 156.4899 0.0 L 0 setlinecap stroke  end �s" tx@Dict begin STP newpath 0.8 SLW 0  setgray  125.19193 62.59596 moveto 125.19193 0.0 L 0 setlinecap stroke  end �w" tx@Dict begin STP newpath 0.8 SLW 0  setgray  62.59596 31.29797 moveto 125.19193 31.29797 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  125.19193 31.29797 moveto 62.59596 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  62.59596 0.0 moveto 62.59596 31.29797 L 0 setlinecap stroke  end �����	color pop��u?E��>�����color push gray 0�i?E��������EG�color push gray 0Figure�UU5:�q�A�6-tiling,�an�8-tiling,�and�a�12-tiling�	color pop���XM~��src:598 SevenTriangles.tex���s��g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 85.35826 0.0 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 42.67912 73.92262 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 73.92262 moveto 85.35826 0.0 L 0 setlinecap stroke  end �k" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 64.01869 36.9613 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 73.92262 moveto 42.67912 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 36.9613 moveto 85.35826 0.0 L 0 setlinecap stroke  end ������	��g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 85.35826 0.0 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 42.67912 73.92262 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 73.92262 moveto 85.35826 0.0 L 0 setlinecap stroke  end �t" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 36.9613 moveto 64.01869 36.9613 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 36.9613 moveto 21.33955 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  64.01869 36.9613 moveto 64.01869 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 73.92262 moveto 42.67912 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 36.9613 moveto 42.67912 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 0.0 moveto 64.01869 36.9613 L 0 setlinecap stroke  end ������;�g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 85.35826 0.0 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 42.67912 73.92262 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 73.92262 moveto 85.35826 0.0 L 0 setlinecap stroke  end �t" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 36.9613 moveto 64.01869 36.9613 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 36.9613 moveto 42.67912 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 0.0 moveto 64.01869 36.9613 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 21.33955 12.31999 L 0 setlinecap stroke  end �u" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 12.31999 moveto 21.33955 36.9613 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  21.33955 12.31999 moveto 42.67912 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 0.0 moveto 64.01869 12.31999 L 0 setlinecap stroke  end �u" tx@Dict begin STP newpath 0.8 SLW 0  setgray  64.01869 12.31999 moveto 64.01869 36.9613 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  64.01869 12.31999 moveto 85.35826 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 24.6413 moveto 42.67912 0.0 L 0 setlinecap stroke  end �t" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 24.6413 moveto 21.33955 36.9613 L 0 setlinecap stroke  end �t" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 24.6413 moveto 64.01869 36.9613 L 0 setlinecap stroke  end �v" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 49.28131 moveto 42.67912 73.92262 L 0 setlinecap stroke  end �u" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 49.28131 moveto 21.33955 36.9613 L 0 setlinecap stroke  end �u" tx@Dict begin STP newpath 0.8 SLW 0  setgray  42.67912 49.28131 moveto 64.01869 36.9613 L 0 setlinecap stroke  end �����	color pop����W�the���problem���of�determining�the�v��|ralues�of��N��l�for�whic��9h�there�exists�some����W�N����-tiling.�r�The�2-pro�A�of�2,giv��9en�here�for�the�sp�A�ecial�case��N����=���7�ma��9y�still�b�A�e�of����Wsome�m�in��9terest,���since�it�uses�only�m�elemen�tary�metho�A�ds�of�of�m�Euclidean�ge-����Wometry��X(including��Ysome�elemen��9tary�trigonometry��:�,���whic�h�could��Xb�A�e�done�b��9y����Wgeometric��metho�A�ds).��T��:�o��tac��9kle�the�next�in�teresting�case,�Ҿ�N����=���11�b�y�these����Wmetho�A�ds�0�w��9ould�require�h�undreds,�7xif�not�thousands,�7wof�pages.�n`Luc�kily��:�,�7xw�e����Wfound�Ta�more�abstract�approac��9h�in�[�1����].����d��src:647 SevenTriangles.texThe�@examples�@of��N����-tilings�giv��9en�ab�A�o�v�e�@are�w�ell-kno�wn.���They�@ha�v�e����Wb�A�een�nGdiscussed,���in�particular,�in�connection�nHwith�\rep-tiles"�[�5����].�'JA�n0\rep-����Wtile"�+ais�+ba�set�of�p�A�oin��9ts��X�߼�in�the�plane�(not�necessarily�just�a�triangle)����Wthat���can���b�A�e�dissected�in��9to��N����congruen�t�sets,��Zeac�h�of���whic�h�is���similar�to����W�S���.�?�An�!'�N����-tiling�in�!&whic��9h�the�tiled�triangle��AB�r�C���is�similar�to�the�triangle����W�T�N��used�as�the�tile�is�a�sp�A�ecial�case�of�this�situation.��That�is�the�case,����Wfor��Nexample,��for�the��n���-=�2����family�and��Othe��n���-=�2���0�+����m���-=�2���family��:�,��but��Onot�for�the����W3-tiling,��M6-tiling,�or��the�12-tiling�exhibited��ab�A�o��9v�e.���Th�us�the��concepts�of����Wan�B��N����-tiling�B�and�rep-tiles�o��9v�erlap,��but�B�neither�subsumes�the�other.���As����Wfar��qas��pI��ha��9v�e�so��qfar�b�A�een�able�to�disco��9v�er,��xthere�is��q(un�til�no�w)��pnot�a����Wsingle�<�publication�<�men��9tioning�the�concept�of�an��N����-tiling�in�general.��yThe����Wpap�A�er��W[�4����]�also�con��9tains�a�diagram��Vsho�wing�the��n���-=�2����family�of�tilings,���but�the����Wproblem���considered���there�is�dieren��9t:�Yone�is�allo��9w�ed���to�cut��N����copies�of����Wthe��ctile��brst,���b�A�efore�assem��9bling�the�pieces�in��9to�a�large�gure,���but�the�large����Wgure��m��9ust�b�A�e��similar�to�the�original�tile.�>The�t�w�o�b�A�o�oks��[�2����]��and�[�3��]�ha��9v�e����Wtan��9talizing�Ttitles,�but�deal�with�other�problems.�������W�2��oL�Denitions�ffand�Notation��阍�W�src:657 SevenTriangles.tex�W��:�e�S�giv��9e�S�a�mathematically�precise�denition�of�\tiling"�and�x�some�ter-����Wminology� and� 	notation.�<�Giv��9en�a�triangle��T�h��and�a�larger�triangle��AB�r�C����,�"�a�����>�color push gray 0����3��Y�	color pop����2�y�������>�color push gray 0�Y�	color pop���?���TQ���>���u�color push gray 0�zQ�����<���#��color push gray 0�Figure�UU6:�q�A�13-tiling�	color pop���i_č�src:625 SevenTriangles.tex���8���g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 204.7499 0.0 L 0 setlinecap stroke  end �m" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 147.88432 85.35826 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  147.88432 85.35826 moveto 204.7499 0.0 L 0 setlinecap stroke  end �s" tx@Dict begin STP newpath 0.8 SLW 0  setgray  147.88432 0.0 moveto 147.88432 85.35826 L 0 setlinecap stroke  end �s" tx@Dict begin STP newpath 0.8 SLW 0  setgray  176.29758 0.0 moveto 176.29758 42.67912 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  49.28131 0.0 moveto 49.28131 28.45232 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  98.56264 0.0 moveto 98.56264 56.90593 L 0 setlinecap stroke  end �w" tx@Dict begin STP newpath 0.8 SLW 0  setgray  98.56264 57.85413 moveto 147.88432 57.85413 L 0 setlinecap stroke  end �w" tx@Dict begin STP newpath 0.8 SLW 0  setgray  49.28131 28.45232 moveto 147.88432 28.45232 L 0 setlinecap stroke  end �x" tx@Dict begin STP newpath 0.8 SLW 0  setgray  147.88432 42.67912 moveto 176.29758 42.67912 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  147.88432 56.90593 moveto 49.28131 0.0 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  147.88432 28.45232 moveto 98.56264 0.0 L 0 setlinecap stroke  end �s" tx@Dict begin STP newpath 0.8 SLW 0  setgray  147.88432 42.67912 moveto 176.29758 0.0 L 0 setlinecap stroke  end �����	color pop��`J��W�\tiling"���of���triangle��AB�r�C�N��b��9y�triangle��T���is�a�list�of�triangles��T�����1��*��;����:�:�:��
�;���T�����;�cmmi6�n�����con-����Wgruen��9t�G�to�G��T�H��,�T�whose�in�teriors�G�are�disjoin�t,�T�and�G�the�closure�of�whose�union����Wis�fHtriangle�fG�AB�r�C����.�LA�f2\strict�v��9ertex"�of�the�tiling�is�a�v��9ertex�of�one�of�the����W�T�����i����that��Mdo�A�es�not��Nlie�on�the�in��9terior�of�an�edge�of�another��T�����j�����.�#\A�� \strict����Wtiling"��is�one��in�whic��9h�no��T�����i��E��has�a�v��9ertex�lying�on�the�in�terior��of�an�edge����Wof�q1another�q2�T�����j�����,��(i.e.�0	ev��9ery�v�ertex�q2is�strict.�0F��:�or�example,��(the�tilings�sho��9wn����Wab�A�o��9v�e��for���N����=���5�and��N��=���13�are��not�strict,���but�all�the�other�tilings�sho��9wn����Wab�A�o��9v�e���are�strict.��}The�letter�\�N����"�will�alw��9a�ys���b�e�used���for�the�n��9um�b�er���of����Wtriangles�~�used�~�in�the�tiling.�YRAn��N����-tiling�of��AB�r�C�"��is�a�tiling�that�uses��N����W�copies�Tof�some�triangle��T�H��.����d��src:666 SevenTriangles.texLet���a�,����b�,���and���c��b�A�e�the�sides�of�triangle��AB�r�C����,���and�angles�����,���x,�,�and��
���b�A�e����Wthe��5angles�opp�A�osite��6sides��a�,���b�,�and��c�,�i.e.�
�the��5in��9terior�angles�at�v��9ertices��A�,����W�B�r��,�[�and�M��C����.���An�M��interior��vertex��in�a�tiling�of��AB�C���is�a�v��9ertex�of�one�of��T�����i�����W�that�߂do�A�es�not�߃lie�on�the�b�oundary�of��AB�r�C����.�
�A��t�b��oundary��vertex�߃�is�a�v��9ertex����Wof�Tone�of�the��T�����i��A��that�lies�on�the�b�A�oundary�of��AB�r�C����.����d��src:672 SevenTriangles.texIn��the�case�of�a�non-strict�tiling,��there�will�b�A�e�a�non-strict�v��9ertex����W�V�8�;�YSso��S�V����lies�on�an�edge�of��T�����j�����,�#Swith��T�����j���.�on�one�side�of�the�edge�and��T�����i�����W�(ha��9ving�9xv�ertex��V�8�)�on�the�other�side.���Consider�the�maximal�line�segmen�t����W�S����extending�
�this�
�edge�whic��9h�is�con�tained�
�in�the�union�of�the�edges�of�the����Wtiling.�*�Since�owthere�ovare�triangles�on�eac��9h�side�of��S���,���there�are�triangles�on����Weac��9h�-,side�of��S��2�at�ev�ery�p�A�oin�t�of��S��1�(since��S��2�cannot�extend�b�A�ey�ond�the����Wb�A�oundary�sof�t�AB�r�C����).�%�Hence�the�length�of��S��y�is�a�sum�of�lengths�of�sides�of����Wtriangles�ɕ�T�����i����in�ɖt��9w�o�dieren�t�w�a�ys�(though�the�summands�ɖma�y�p�A�ossibly�b�e����Wthe��#same��$n��9um�b�A�ers�in�a��$dieren�t�order).�
�Let��$us�assume�for�the�momen��9t����Wthat��Othe�summands��Pare�not�the�same�n��9um�b�A�ers.�nThen��Oit�follo�ws��Othat�some����Wlinear�Trelation�of�the�form���K��Ʉ%�pa�8�+��q�R�b��+��r�A�c����=�0���J��W�src:681 SevenTriangles.texholds,�;�with�4�p�,��q�R��,�and��r�u��in��9tegers�not�4all�zero�(one�of�whic�h�m�ust�of�course����Wb�A�e��negativ��9e),�ʬand�the�sum�of�the�absolute�v��|ralues�of��p�,�ʫ�q�R��,�and��r����is�less�than����Wor�Tequal�to��N����,�since�there�are�no�more�than��N�8�triangles.����d��src:684 SevenTriangles.texBy�Tthe�la��9w�of�sines�w�e�ha�v�e���a������-ꍒ��d�a������H�fe�d�'m���sin���������֑L�=�����-ꍑ��b�����H�fe���'m���sin�����������=�����-ꍑ
ܹ�c�����H�fe0��'m���sin����
�������ꍑW�src:686 SevenTriangles.tex�Up�Tto�similarit��9y�then�w�e�ma�y�assume������	/�a�������=�������!nsin���t��������>�color push gray 0����4��Y�	color pop����Q9�y�������>�color push gray 0�Y�	color pop���?����������b�������=�������!nsin���t����������c�������=�������!nsin���t�
�������W�src:692 SevenTriangles.tex�Since�T�
��=�����`���8�(����+���x,�)�Tw��9e�ha�v�e��sin��p�(�
����)���=��sin��
�"(����+�8��x,�),�Tso�������p�����sin��p�����+�8�q���T�sin���Z���d�+��r���q�sin��
'�(���+���x,�)���=�0�:����d��src:695 SevenTriangles.tex�If����S���is�a�maximal�segmen��9t�con�taining�a�non-strict�v�ertex,���then�there����Wwill��b�A�e��in��9tegers��n��and��m��suc��9h�that��n��triangles�ha��9v�e��a�side�con��9tained�in��S����W�and�F�lie�on�one�side�of��S���,�R�and��m��triangles�ha��9v�e�F�a�side�in��S�Ʌ�and�lie�on�the����Wother�j�side�of��S���.��In�that�case�w��9e�sa�y��S���is�j�of�t�yp�A�e��m����:��n�.��F��:�or�j�example,��Fig.�3����Wsho��9ws�Va�5-tiling�with�a�maximal�Vsegmen�t�of�t�yp�A�e�1���:�2.�ܰThis�Vdenition�do�es����Wnot�trequire�tthat�the�lengths�of�the�sub�A�divisions�of�the�maximal�segmen��9t����Wall�Tb�A�e�the�same�(as�they�are�in�Fig.�p3).�������W�3��oL�2-tilings,�ff3-tilings�and�4-tilings��阍�W�src:776 SevenTriangles.tex�In���this���section,��<w��9e�w�arm�up���b�y�c�haracterizing�2-tilings,��<3-tilings���and�4-����Wtilings.��Not�נonly�will�these�results�b�A�e�used�later,��but�the�ideas�in��9tro�duced����Win�Tthe�pro�A�ofs�will�also�b�e�used�later.������R�color push gray 0��Lemma���1�	color pop������src:781 SevenTriangles.tex�If,��in��3a��4tiling,��P�:�is�a��3b��oundary�vertex�(or�a�non-strict�interior����Wvertex)�~�and�only�one�interior�e��dge�emanates�fr��om��P�H��,��0then�b�oth�angles�at����W�P���ar��e�N<right�angles�and��
�T�=�����R�=�2�.����W�src:785 SevenTriangles.texPr��o�of.����If���either�the���t��9w�o���angles�at��P�כ�are�dieren��9t,��then�their�sum�is�less����Wthan�����R��,��$since�the���sum�of�all�three�angles�is���R��.�|�Therefore�the�t��9w�o���angles����Ware�4mthe�4lsame.�y�But�2���a<�ř�$��+�"���=�<�Ś���and�2��=�<����+�"��
�I'<��R��.�y�Therefore�b�A�oth����Wangles�Tare��
����.�pBut�then�2�
��=�����R��,�so��
��=���R�=�2.������R�color push gray 0��Theorem���1�	color pop�������src:793 SevenTriangles.tex�If�itriangle�i�AB�r�C���is�2-tile��d�by��T�H��,�o�then��AB�r�C���is�isosc��eles�and����Wthe�N<tiling�divides�it�into�two�right�triangles�by�me��ans�of�an�altitude.����W�src:797 SevenTriangles.texPr��o�of.��"�The���t��9w�o���triangles��T�����1���7�and��T�����2���ha��9v�e���a�total���of�2��:�angles,�؞of�whic��9h����W��l�are�oaccoun��9ted�nfor�b�y�othe�v�ertices�of�o�AB�r�C����.�(�An�in�terior�v�ertex�o(strict�or����Wnot)��w��9ould��require�there�to�b�A�e�three�triangles.��Hence�there�is�exactly�one����Wmore�ݽv��9ertex,���and�it�ݼis�a�b�A�oundary�v�ertex.�	�Call�that�ݼv�ertex��P�H��.�	�Since�there����Ware��only��t��9w�o�triangles,��1only�one�in�terior��edge�emanates�from��P�H��,��1and�its����Wother�ݸend�m��9ust�b�A�e�at�the�opp�osite�ݷcorner�of�triangle��AB�r�C����.�u�Relab�eling����Wif��'necessary��:�,��[w��9e�can�assume�this�corner�is��B���and��P��
�lies��&on��AB�r��.���By�the����Wlemma,��C�
�T�=�����R�=�2�Z�and�Z�the�angles�at��P����are�righ��9t�angles.��TThen��AB�{�=����B�r�C�5��=��c����W�since��Ythese�sides�are��Zopp�A�osite�the�righ��9t�angles�of��T�����1����and��T�����2����resp�ectiv��9ely��:�.����WHence���triangle����AB�r�P�D��is�congruen��9t�to�triangle��C���B�r�P�D��and�the�tiling�is�as����Wdescrib�A�ed�Tin�the�theorem.�pThat�completes�the�pro�of.����d��src:806 SevenTriangles.texW��:�e�)Qcould�)Rha��9v�e�reac�hed�the�)Rconclusion�that�just�one�in��9terior�edge�em-����Wanates�yfrom�z�P�e\�in�another�w��9a�y��:�,�^Cwhic�h�yseems�o��9v�erly�ycomplicated�in�this����Wexample,��*but���will�b�A�e���useful�b�elo��9w.��The���triangles��T�����1����and��T�����2����ha�v�e�together����Wsix��b�A�oundary��segmen��9ts.�p[F��:�our�of�these�o�A�ccur�on�the�b�A�oundary�of��AB�r�C����,����Wand���the�other���remaining�b�A�oundary�segmen��9ts�m�ust�suce���to�coun�t�eac�h����Win��9terior���edge�from�b�A�oth�sides.��nIn�this�case�there�are�just�t�w�o�remaining����W(b�A�ecause���6�]����4�Y=�2)�and�hence���there�is�exactly�one�in��9terior�edge,���whose�����>�color push gray 0����5��Y�	color pop����fW�y�������>�color push gray 0�Y�	color pop���?������W�t��9w�o�EZsides�accoun��9t�EYfor�these�t�w�o�b�A�oundary�segmen�ts.���In�general�EYin�an��N����-����Wtiling���there�are��N������radians�to�accoun��9t�for,��of�whic�h�����are�in�the�corners�of����Wtriangle�z�AB�r�C����,��and�zthe�rest�are�distributed�b�A�et��9w�een�zb�oundary�zand�in��9terior����Wv��9ertices.�ԱIf�R�there�R�are��k��M�b�A�oundary�v�ertices�then�there�R�are��k�z��+�7*3�b�A�oundary����Wsegmen��9ts��Hon��Gthe�b�A�oundary�of��AB�r�C����,���lea��9ving�3�N������k�+���� �3�to�b�A�e�accoun��9ted�for����Wb��9y�
coun�ting�eac�h�side�of�eac�h�in�terior�edge.�XIn�a�strict�tiling,�
�the�n�um�b�A�er����Wof���in��9terior���edges�will�th��9us�b�A�e�half�of�3�N��)���E�k�2����3,���but���in�a�non-strict�tiling,���a����Wmore���detailed���accoun��9ting�m�ust�b�A�e���made.�
�In�the�next�pro�A�of,��w��9e�will�apply����Wthis�Ttec��9hnique�to�the�case��N����=���3.���%����R�color push gray 0��Theorem���2�	color pop�������src:820 SevenTriangles.tex�If���triangle����AB�r�C�\��is�3-tile��d�by��T�H��,��then�either�(i)��AB�r�C�\��is����We��quilater�al���and�the�tiling�c��onsists�in�c�onne�cting�the�c�enter�of��AB�r�C�2��to�its����Wvertic��es,�`or���(ii)����AB�r�C�[��is�a�30-60-90�triangle,�`and�ther��e�is�no�interior����Wvertex�8of�9the�tiling;��the�shar��e�d�side�8of�two�of�the��T�����i��0��is�p��erp�endicular�to�8the����Whyp��otenuse���of����AB�r�C�r��at�its�midp��oint��P�H��,��\and�me�ets���side��b��at��Q�,��\say,��[and�the����Wother��interior�e��dge��c�onne�cts���Q��to�the�vertex��B����(wher��e�the�angle�of��AB�r�C����W�is�N<��R�=�3�).�@Se��e�Fig.�1.���$��W�src:827 SevenTriangles.texPr��o�of�.�65Supp�A�ose�s@�AB�r�C�(�is�3-tiled�b��9y��T�����1��*��,����T�����2���,���and�s@�T�����3���.�64First�sAw�e�s@supp�A�ose�the����Wtiling�JXis�JYstrict.��}The�total�of�the�angles�in�the�tiling�is�3��R��,�W�since�there�are����Wthree�F\copies�of�F]�T�H��.���The�total�angle�accoun��9ted�for�b�y�the�F]v�ertices�of��AB�r�C����W�is��6��R��.��Eac��9h�strict�in�terior��5v�ertex�accoun�ts�for��52�����and�eac�h��5b�A�oundary�v�ertex����Wfor�����R��.�ZTh��9us�there���are�only�t��9w�o���p�A�ossibilities:�_�one�in�terior���v�ertex�and���no����Wb�A�oundary�Tv��9ertices,�or�t�w�o�b�A�oundary�v�ertices�and�no�in�terior�v�ertex.����d��src:835 SevenTriangles.texFirst�7�assume�7�that�there�is�one�in��9terior�v�ertex�7�and�no�b�A�oundary�v��9ertices.����WSince�pthere�pare�no�b�A�oundary�v��9ertices,�� three�of�the�nine�b�A�oundary�segmen��9ts����Wof�.�the��T�����i��[=�are�on�the�b�A�oundary�of�.��AB�r�C����,�u'and�the�other�six�are�double-����Wcoun��9ted���as�the�t�w�o�sides���of�three�in�terior�edges.���Since�at�least�three�edges����Wm��9ust���emanate���from�an�in�terior���v�ertex,��all�three���edges�do���emanate�from����Wthe�u�one�u�in��9terior�v�ertex�u��P�H��.��DSince�there�u�are�no�b�A�oundary�v��9ertices,���they�m�ust����Wterminate�woin�wpthe�three�v��9ertices��A�,����B�r��,���and��C����.�B�That�is,�at�least�wpthe�tiling����Whas���the�top�A�ology���of�the�tiling�in�(i).��The�three�angles�at��P�Cs�m��9ust�all�b�A�e��
����,����Wsince��6an��9y��5other�sum�of�three�angles�c��9hosen�from�����,�����x,�,�and��
���is�at��6most����W���d�+�82�
����,�Twhic��9h�is�less�than�2��g��b�A�ecause��6܍����,����d�+�82�
������@�=������F���d�+�8����+�(�
���������)�+��
���������@�=������F��`��+�8�
���������������@<������F�2�����6ۍ�W�src:848 SevenTriangles.tex�Hence���3�
�/\�=���2��R��,���so��
�/[�=�2��R�=�3.�Hence�the����c��sides�of�all�three��T�����i����are�the����Wfaces��of�triangle��AB�r�C����,���whic��9h�is�th�us�equilateral.��No�w�let��AP�ک�=����a�;��{then�in����Wtriangle��5�AP�H�C����,��ow��9e�ha�v�e��6�P�H�C�5��=����b�;���hence�in�triangle��C���P�H�B�r��,��ow��9e�ha�v�e��P�H�B�{�=����a�;����Whence���in���triangle��P�H�B�r�A��w��9e�ha�v�e����AP�ک�=����b�.��1Hence��AP��z�is�equal�to�b�A�oth��a��and����W�b�,�+�so�'=�a����=��b��and�'<the��T�����i��S��are�isosceles.�R*Hence�the�tiling�is�the�one�describ�A�ed����Win�T(i)�of�the�theorem.����d��src:855 SevenTriangles.texNext�i�assume�i�that�there�are�t��9w�o�b�A�oundary�i�v�ertices��P����and��Q�i��(and�hence����Wno�e�in��9terior�e�v�ertex).�
�Then�there�are�e�v�e�b�A�oundary�segmen�ts,�y�i.e.�
�sides�of����Wcopies��Dof��T�'�lying�on�the�b�A�oundary�of��E�AB�r�C����.�V@Since�there�are�only�three����Wtriangles,���t��9w�o���of�the�triangles���m��9ust�accoun�t�for�t�w�o�sides���eac�h,���i.e.��Ot�w�o����Wof���the�angles�of����AB�r�C�N��are�not�\split",��.i.e.���are�not�shared�b��9y�more�than����Wone�v��T�����i��,r�.�AHence�v�eac��9h��T�����i���R�is�similar�to�triangle��AB�r�C����.�AOf�the�9�sides�of�the�����>�color push gray 0����6��Y�	color pop����xy�y�������>�color push gray 0�Y�	color pop���?������W�three����T�����i��,r�,��v��9e�o�A�ccur�on�the�b�oundary�of��AB�r�C����,��and�the�other�4�o�ccur�in����Wthe��oin��9terior.��Since��neac�h�in�terior�side�is�coun�ted�t�wice,��5as�a�b�A�oundary�of����Wthe��jtriangles�on��keither�side,���there�m��9ust�b�A�e�exactly�t�w�o�in�terior��kedges.��wOne����Wof�\�these�\�in��9terior�edges�m��9ust�connect��P����and��Q�,�n�b�A�ecause�if�not,�then�b�A�oth����Win��9terior�(�edges�(�w�ould�ha�v�e�to�(�connect��P�q|�or��Q��to�the�opp�A�osite�v��9ertex.�V>But����Wif��hone�edge�connects�(sa��9y)��Q��to�the�opp�A�osite�v�ertex,���then�the�edge�from��P����W�is�k_blo�A�c��9k�ed�k^from�reac�hing�the�opp�A�osite�k^v�ertex,���and�vice-v�ersa,���if�one�k_edge����Wconnects�ƽ�P���to�the�opp�A�osite�v��9ertex,��the�other�edge�cannot�connect��Q��to����Wthe�&-opp�A�osite�&,v��9ertex.�N�Hence�it�cannot�b�A�e�that�b�A�oth�in��9terior�edges�connect����W�P��<�or�vX�Q�vY�to�the�opp�A�osite�v��9ertex.�?~The�only�other�p�A�ossibilit��9y�is�that�one�of����Wthese���edges�connects��P����to��Q�.���The���other�in��9terior�edge�m�ust�connect�one�of����W�P�B��or����Q��to�the�opp�A�osite���v��9ertex�of��AB�r�C����,��ewhic�h�m�ust�b�A�e�split.�LThat�means����Wthat���one�of��P�
��or��Q��(b��9y�relab�A�eling�w�e�can�assume�it���is��P�H��)�has�only�one����Win��9terior�?=edge�?<emanating�from�it.��*That�implies�that��
����is�a�righ��9t�angle,�I�b�y����WLemma�<1.���Changing�the�lab�A�els��A�,�E��B�r��,�E�and��C���if�necessary��:�,�w��9e�can�assume����Wthat��p�P�/S�lies�on��AB�r��,����Q��lies�on��B�C����,���and��q�QA��and��QP�/S�are�the�in��9terior�edges.����WAngles�gL�QP�H�B���and��QP�A��are�righ��9t�angles,�{�and�triangle��AQP��/�is�congruen�t����Wto�"]triangle��QP�H�B�r��.�C�Sides��AQ��and��QB���are�opp�A�osite�"\the�righ��9t�angle�and����Whence��are�equal.�7�Hence��AP�)�=��F�P�H�B�;��and��P���is�the�midp�A�oin��9t�of��AB�r��.�The����Wangle��at��B�{��is��not�split.�SSince�triangle��AB�r�C����is�similar�to�eac��9h�triangle��T�����i��,r�,����Wbut�k�its�k�area�is�3�times�larger,���the�similarit��9y�factor�is������?�p���!@���?�aH���u���3�����>.� <Let��T�����1�����and��T�����2�����W�b�A�e���the���t��9w�o�triangles���sharing�side��P�H�Q�,���with��T�����1�����=�f�QP�A��and����T�����2�����=�f�QP�B�r��.����WThen���T�����3����shares�side��AQ�,��whic��9h�is�side���c��in�triangle��T�����1��*��,�so�the�third�v��9ertex����W�C� z�of�|��AB�r�C����,��awhic��9h�is�|�also�the�v��9ertex�of��T�����3���9�opp�A�osite��AQ�,��am��9ust�b�e�|�the����Wrigh��9t-angled�R�v�ertex�of�R��AB�r�C����.���No�w�triangle�R��C���AB��z�is�similar�to�triangle����W�P�H�QB�r��,�JCsince�?�they�ha��9v�e�?�the�?�same�angle�at��B��a�and�righ��9t�angles�at��C���and��P����W�resp�A�ectiv��9ely��:�.��Hence��7�AB���and��QB��are��6corresp�A�onding�sides.��Their�ratio�is����Wtherefore���]���?�p���`���?�aH���u���3�����^,�n�i.e.���AB�|�=���	Q���?�p���
�����?�aH���u���3����^��QB�r��.�But�]since��AB�|�=�	Q�AP����+�>	�P�H�B��=�	R2�AV�8�,�w��9e����Wha��9v�e�o2�P�H�B���=���'b���?�p���
ܴ���?�aH���u���3����|��QB�r��.�)�Hence�angle�o�B���=�'a��R�=�6�and�angle��P�H�QB���=�'a��R�=�3,���and����Wthe��tiling��is�as�describ�A�ed�in�(ii)�of�the�theorem.��That�completes�the�pro�A�of����Win�Tcase�of�a�strict�tiling.����d��src:887 SevenTriangles.texNo��9w�Ssupp�A�ose�the�tiling�is�non-strict.�lSince�only�three�triangles�are����Win��9v�olv�ed,��'the��0only��/p�A�ossible�t��9yp�e�of�non-strict�v��9ertex�is��/the�t�yp�A�e�w�e�shall����Wcall��h2���:�1�b�A�elo��9w,�ȗwhere�one�side�of�(sa�y)��g�T�����1����is�matc�hed�b�y�t�w�o�sides,�ȗone�of����W�T�����2��ƺ�and��one�of��T�����3��*��.��There�cannot�b�A�e��t��9w�o��suc�h�v�ertices�as�the�three�triangles����Wwill�u�ha��9v�e�only�u�this�one�side�of��T�����1���`�in�common,���and�if�the�sides�of��T�����2���_�and����W�T�����3����that��mtouc��9h��ldo�not�ha�v�e�the��lsame�length,���a�triangle��AB�r�C�.T�will�not�b�A�e����Wformed.���Hence,��2of��ithe�3���in�total�angles,����is�accoun��9ted�for�at�the�in�terior����Wv��9ertex����P�H��,�"�and���?|�is���accoun�ted�for�b�y�the�v�ertices���of��AB�r�C����,�"�lea�ving���?|�to����Wb�A�e�E�accoun��9ted�for�b�y�E�a�single�b�A�oundary�v�ertex��Q�.��With�one�b�A�oundary����Wv��9ertex�O�there�are�4�O�b�A�oundary�segmen�ts�on�the�b�A�oundary�of��AB�r�C����,�^�lea�ving����W3�����3�����4���=�5�~in�the�in��9terior�(coun�ting�eac�h�side�of�eac�h�in�terior�segmen�t).����WThree��of���those�are�the�three�sides�that�lie�on�the�maximal�segmen��9t�of�the����Wnon-strict��v��9ertex��P�H��.�The�other�t�w�o�are�the�t�w�o��sides�of�one�more�in�terior����Wsegmen��9t��with�an�endp�A�oin�t�at��P�H��.�
�Since��there�is�only�one�b�A�oundary�v�ertex,����Wthe��.three��/endp�A�oin��9ts�of�the�in��9terior�segmen�ts��.m�ust�end��.at��Q��and�at�t��9w�o����Wcorners��yof�the�triangle.���The�maximal��xsegmen��9t�m�ust�end�at��Q��and�one����Wcorner,�d�whic��9h�T�w�e�T�ma�y�lab�A�el��A�,�d�and�the�T�other�in�terior�segmen�t�runs�from����W�P����to�J�another�J�corner,�W�sa��9y��B�r��.��OSince�only�t��9w�o�J�triangles�share�v��9ertex��Q�,�W�w�e����Wha��9v�e���
�&R�=�����R�=�2��b�y�Lemma�1.�;But�no�w�triangle���QP�H�B��;�has�t�w�o�righ�t�angles,�����>�color push gray 0����7��Y�	color pop������y�������>�color push gray 0�Y�	color pop���?������W�at�T�Q��and��P�H��.�pThat�con��9tradiction�completes�the�pro�A�of.������R�color push gray 0��Theorem���3�	color pop�������src:906 SevenTriangles.tex�If��triangle���AB�r�C���is�4-tile��d�by��T�H��,���then�(a)�ther��e�is�no�interior����Wvertex,�mand�3�(b)��T�|��is�a�30-60-90�triangle,�and�the�tiling�is�one�of�those����Wshown��Lin��MFig.�W43�(or�a�r��e
e�ction�of��Lthese),�‘or��T��/�c��an�b�e��Lany�triangle�and����Wthe�N<tiling�is�one�of�the��n���-=�2��x��family�as�il�x�lustr��ate�d�N<in�Fig.�@2.����W�src:911 SevenTriangles.texPr��o�of�.�,�First��fsupp�A�ose��gthe�tiling�is�strict.�,�The�four�triangles�ha��9v�e�angles����Wtotaling���4��R��.��1The�v��9ertices���of��AB�r�C�4�accoun�t�for����B�of�this,��$and�the�remaining����W3���U�m��9ust���b�A�e�accoun�ted�for.��vThere�are�just�t�w�o�p�A�ossibilities:�!one�in�terior����Wv��9ertex��and��one�b�A�oundary�v��9ertex,�H�or�no�in��9terior�v�ertices��and�three�b�A�oundary����Wv��9ertices.����d��src:916 SevenTriangles.texFirst�u�assume�u�there�is�one�in��9terior�v�ertex�u��P����and�one�b�A�oundary�v��9ertex����W�Q�.��=Then���there���are�four�b�A�oundary�segmen��9ts�and�(12�g���4)�=�2�o�=�4���in��9terior����Wedges.�"Then�l�these�l�four�edges�m��9ust�emanate�from��P��o�and�go�to��A�,��Z�B�r��,��C����,����Wand�4L�P�H��.�yWBy�4Kthe�lemma,�|	the�angle�at��Q��is�a�righ��9t�angle�and��
���=�p
��R�=�2.����WHence� �all�four� �angles�at��P�i��m��9ust�b�A�e�righ��9t�angles.�>�Then�triangle��AP�H�Q��has����Wt��9w�o�Jrigh�t�J�angles,�W�con�tradiction.���That�disp�A�oses�of�J�the�case�of�one�in��9terior����Wv��9ertex�Tand�one�b�A�oundary�v�ertex.����d��src:922 SevenTriangles.texNext��:assume��9there�are�three�b�A�oundary�v��9ertices�and�no�in��9terior�v�ertex.����WThen�Bthere�Care�six�b�A�oundary�segmen��9ts�and�(12����6)�=�2���=�3�in��9terior�Bedges.����d��src:925 SevenTriangles.texFirst�q�assume�q�that�one�of�the�in��9terior�edges�terminates�in��A�,����B�r��,���or��C����W�(splitting���the�angle�there).��gThen�there�are�not���enough�edges�to�pro��9vide����Wt��9w�o�%edges�$at�eac��9h�b�A�oundary�v��9ertex,�.so�one�b�A�oundary�v��9ertex�has�only�one����Wedge��wterminating��vthere.���Hence�b��9y�the�lemma,����
����=�vU��R�=�2.���Lab�A�el�the�split����Wv��9ertex��B����and�let��Q��b�A�e�the�in�terior�v�ertex�at�the�other�end�of�the�in��9terior����Wedge�J�emanating�from��B�r��.���Then�either�triangles��AB�Q��and��C���B�Q��are�b�A�oth����W2-tiled,���or���one���is�3-tiled�and�the�other�is�congruen��9t�to�the�tile��T�H��.�!�First����Wassume�q>�AB�r�Q��is�q?3-tiled.��There�are�t��9w�o�q>p�A�ossible�3-tilings;���in�one�case,��angle����W�AQB����is�2��R�=�3,�_wand�1�the�fourth�triangle��T�����3��\��can�con��9tribute�only�an�angle�of���R�=�6����Wat�K�eac��9h�v�ertex,�Y�not�enough�to�mak�e������and�remo�v�e�a�v�ertex.��So�this�case����Wis��imp�A�ossible.���In�the�other�p�ossible�3-tiling,��p�AB�r�Q���is�a�30-60-90�triangle����Wsimilar���to����T�H��,���and�the�three�sides�of��AB�r�C�^u�are��a�|^�+��c�,���2�b�,�and����b�.�Neither����W�a����+��c��nor�2�b��can�b�A�e�	a�side�of��T�H��,�Q5so�w��9e�m�ust�ha�v�e��QB����=�6��b�,�Q5and�angle����W�C�8�=��4��R�=�3.� �W��:�e��then�necessarily�ha��9v�e��the��third�tiling�sho��9w�in�Fig.�3�(or�its����Wre
ection).�Next�h6assume�that��AB�r�Q��is�h52-tiled.�Then��AQ���=��QB����and�h6�QP����W�is��an�altitude��of�triangle��AQB�r��.���The�third�in��9terior�v�ertex��R��N�m�ust�lie�on����W�QC���or�@�on�@��B�r�C����.���First�assume��R�Q�lies�on��B�C����.���Then�@�the�third�in��9terior�edge����Wis����QR�>�,��and�b�A�oth���angles�at��R��	�are�righ��9t�angles�b��9y�the�lemma.�	�W��:�e�therefore����Wha��9v�e���the���rst�tiling�sho��9wn�in�Fig.�s�3.�Next���assume��R���lies�on��AC����.�s�Then����Wthe���third���in��9terior�edge�m�ust�b�A�e����B�r�R�>�,��^and�angle��QR�>B�U�m��9ust�b�A�e�a�righ��9t�angle����Wb��9y��Xthe�lemma.�qThen�triangle��P�H�QB�m
�is�congruen�t��Wto�triangle��R�>QB�m
�and�w�e����Wha��9v�e�G`the�third�tiling�in�Fig.���3.�This�G`disp�A�oses�of�the�case�in�whic��9h�one�of����Wthe�Tin��9terior�edges�terminates�in�a�v�ertex�of��AB�r�C����.����d��src:944 SevenTriangles.texNo��9w�?6the�?5three�in�terior�?5edges�terminate�only�in�the�three�b�A�oundary����Wv��9ertices.�cIt��+follo�ws��,that�triangle��AB�r�C���is�similar�to�triangle��T�H��;��9since�there����Ware��four��triangles,�#�the�similarit��9y�factor�is�2:�� triangle��AB�r�C����is�t��9wice�the����Wsize�a�of��T�H��,�t�and�the�a�same�shap�A�e.�-If�t��9w�o�a�b�A�oundary�v��9ertices�lie�on�the�same����Wside�oof�o�AB�r�C����,���three�in��9terior�edges�cannot�exist.�)�Therefore�one�b�A�oundary����Wv��9ertex��lies��on�eac�h�side��of��AB�r�C����.�
�Lab�A�el�them�so�that��P�)��lies�on��AB�r��,��8�Q��lies����Won���B�r�C����,��xand��R��H�lies�on��AC��.��Then��
the�three�in��9terior�edges�form�triangle�����>�color push gray 0����8��Y�	color pop����	�p�y�������>�color push gray 0�Y�	color pop���?������W�P�H�QR�>�.���Let���R�Q�n��=�n��a�,��2�P�Q��=�n��b�,��2and���R�>Q��=��c�.���Then��if��C���Q��=��a�,��3it��follo��9ws����Wthat�7I�QB�=�=��^�a�7H�and��AR�ڛ�=��R�>C�nE�=��b�,�?�and�w��9e�7Hha�v�e�7Ian��n���-=�2��*��-family�tiling.��MIf,�?�on����Wthe�o[other�o\hand,����C���Q�'��6�=�'��a�,�then�since�o[�R�>Q�'��=�'��c��w��9e�m�ust�o[ha�v�e��C���Q�'��=�'��b�o[�and����W�b����6�=��a�.���But�D�then��R�>C����=����a��and�hence�D��AR��&�=��a�.���But�b��9y�D�denition��R�>P�)��=��a�,����Wand��b��9y��the�similarit�y�of���AB�r�C����to��T�H��,��]w�e��ha�v�e��AB�{�=���2�c��and��hence��AP�ک�=��c�.����WHence�c�a���=���b�,�=�con��9tradiction.��This�completes�the�pro�A�of�in�dthe�case�of�a����Wstrict�Ttiling.����d��src:956 SevenTriangles.texNo��9w�u�assume�that�there�is�a�single�non-strict�v�ertex��P�H��.�=�If�this�v�ertex����Wis��of�t��9yp�A�e�3�$x:�1��then�ev�ery�one�of�the�four�triangles�shares�a�side�with����Wthe�H�maximal�H�segmen��9t��S���.��]There�are�no�more�triangles�that�can�share�the����Wt��9w�o�FFin�terior�FEv�ertices�on��S���,�R�so�FEonly�one�edge�emanates�from�eac��9h�of�these����Wv��9ertices���on�the�side�b�A�ounding�three�triangles.�?By�the�lemma�the�angles�at����Wthese��v��9ertices��are�righ�t��angles.�[But�then�the�middle�of�the�three�triangles����Whas��t��9w�o�righ�t�angles,�"con�tradiction.�;Hence�the��non-strict�v�ertex�do�A�es�not����Wha��9v�e�-{t�yp�A�e�3�d�:�1.�d�If�-|it�has�t��9yp�A�e�2�:�2,�s�then�similarly�-|the�angles�at�the����Win��9terior�%�v�ertices�%�are�righ�t�angles,�)�so�%�the�union�of�the�four�copies�of��T�nv�has����Wfour��qv��9ertices��rand�cannot�b�A�e�a�triangle��AB�r�C����.��Therefore�the�t��9yp�A�e�of�the����Wnon-strict�Tv��9ertex�m�ust�b�A�e�2���:�1.����d��src:966 SevenTriangles.texThe���non-strict���v��9ertex�accoun�ts�for�����x�of�the�4��x�angles�of�the��T�����i��,r�,��land����Wsince�����F��is�accoun��9ted�for���b�y�the�corners�of��AB�r�C����,�+�that�lea�v�es�either�one����Wstrict�&�in��9terior�&�v�ertex�and�no�&�b�A�oundary�v�ertices,�*�or�t�w�o�&�b�A�oundary�v�ertices����Wand�Tno�strict�in��9terior�v�ertices.����d��src:971 SevenTriangles.texFirst�+�assume�+�there�is�one�strict�in��9terior�v�ertex��Q��and�+�no�b�A�oundary����Wv��9ertices.��RThe�H�maximal�H�segmen�t�m�ust�run�H�from�a�v��9ertex�(sa�y��B�r��)�H�to��Q�,�q�since����Wit��cannot��run�to�another�v��9ertex�of��AB�r�C����.��UThe�other�edges�emanating�from����W�Q�zD�m��9ust�b�A�e�zC�AQ��and��B�r�Q�.�K?There�are�3�b�oundary�segmen��9ts�zCand�9�double-����Wcoun��9ted�`�in�terior�`�edges.���The�maximal�segmen��9t,�s�since�it�is�of�t��9yp�A�e�(2��:�1),����Wcon��9tains�*3�*of�these�edges,�/Mand�the�other�6�corresp�A�ond�to�three�additional����Win��9terior��Kedges.���These�are��L�AQ�,�ā�C���Q�,�Āand�the��Kother�edge�emanating�from��P�H��.����WThe��endp�A�oin��9t�of��that�edge�m�ust�b�A�e��a�v�ertex�of��AB�r�C����,��Xwhic�h�b�y�relab�A�eling����Ww��9e��ccan��bassume�is��C����.�� Then�b��9y�the�lemma,���the�angles�at��P��F�are�righ��9t�angles,����Wand�b��
����=�l��R�=�2.�But�only�b�three�angles�meet�at��Q�,�u�so�one�of�them�m��9ust�b�A�e����Wgreater�|than���R�=�2.���This�con��9tradiction�}disp�A�oses�of�the�case�of�one�strict����Win��9terior�Tv�ertex.����d��src:981 SevenTriangles.texTherefore��the��second�case�m��9ust�hold:�there�are�t��9w�o�b�A�oundary��v�ertices����Wand��uno��vstrict�in��9terior�v�ertices.���Then��vthere�are�v�e�b�A�oundary��vsegmen�ts�and����Wsev��9en��5double-coun�ted��6in�terior�segmen�ts,��<of�whic�h��6three�lie�on�the�maximal����Wsegmen��9t,�h�so�Xthere�are�Xt�w�o�additional�in�terior�segmen�ts,�h�one�of�Xwhic�h�has����Wone��3end�at�the��4non-strict�v��9ertex��P�H��.�i
That�mak�es��4v�e�ends�of�in�terior����Wsegmen��9ts�k(t�w�o�lof�the�maximal�segmen��9t,�rand�three�of�the�four�ends�of�the����Wt��9w�o��additional��in�terior�segmen�ts)��that�m�ust��terminate�at�t��9w�o�b�A�oundary����Wv��9ertices�Oand/or�the�Nthree�corners�of��AB�r�C����.�:`The�maximal�segmen��9t�cannot����Wconnect�?�t��9w�o�?�v�ertices�of�?��AB�r�C����,��Mso�it�m��9ust�ha�v�e�?�one�end�at�a�b�A�oundary����Wv��9ertex�9��Q�.��jThe�other�end�9�of�the�maximal�segmen��9t�is�either�at�a�v��9ertex�of����W�AB�r�C��<�or�Tat�the�other�b�A�oundary�v��9ertex��R�>�.����d��src:990 SevenTriangles.texAssume���rst���that�the�maximal�segmen��9t�connects��Q��to�a�v��9ertex�of����W�AB�r�C����.�F9Relab�A�eling,��hw��9e�x�can�assume�it�is�x�v�ertex��B�r��,��hand�the��Q��lies�on��AC����,����Wand���triangle��C���QB�'��is���congruen��9t�to��T����while�triangle��AQB��is���3-tiled.��2By����WTheorem���2,��,it�follo��9ws�that��AQB�lW�is�a�righ�t�triangle�or�is�equilateral.�5First����Wassume�{y�AQB��/�is�equilateral.�N�Then�angle��C���QB��is���R�=�6�and�{zangle��AQC����=�����>�color push gray 0����9��Y�	color pop����
���y�������>�color push gray 0�Y�	color pop���?������W�5��R�=�6,�D�not�;a�straigh��9t�;angle,�D�con�tradiction.���Hence��AQB����is�not�;equilateral.����WHence��=the�other��<case�holds,�Ǩnamely�that��AQB�&��is�a�30-60-90�triangle,�Ǩtiled����Was���in���Fig.�e1.�eSince��QB����is�a�single�side�of��T���in�triangle��QB�r�C����,��Athe�tiling����Wof�e��AQB�ح�m��9ust�b�A�e�orien�ted�e�so�that��QB�ح�is�the�smallest�side�of��QB�r�A�.�\The����Wtiling�Tthen�m��9ust�b�A�e�the�third�one�sho�wn�in�Fig.�p3,�or�its�re
ection.����d��src:999 SevenTriangles.texThe��only�remaining��case�is�that�the�maximal�segmen��9t�connects��Q��to�the����Wother�O�b�A�oundary�O�v��9ertex��R�>�.��Relab�eling,�^w��9e�O�can�assume�that��Q��lies�on��AC����W�and�xF�P��*�lies�on�xG�AB�r��.��Assume�rst�that�an�in��9terior�segmen�t�xFemanates�from��P����W�on��zthe��ysame�side�of��P�H�Q��as��A�.�}Then�its�other�endp�A�oin��9t�m�ust��zb�A�e��A�.�}Hence����Wthe���angles���it�mak��9es�at��P�B��are�righ�t�angles�and����QP�W��=���P�H�R�>�.��The�second����Win��9terior��segmen�t��cannot�ha�v�e�endp�A�oin�ts��at�b�A�oth��Q��and��R�>�,�(�so�at�one�of����Wthose���p�A�oin��9ts,��only�the�maximal�segmen�t�meets�the���b�A�oundary�of��AB�r�C����.��,By����Wthe��lemma�then,���angle��AQR��V�or�angle���B�r�QR��is�a�righ��9t�angle,���con�tradiction,����Wsince�x<that�x;w��9ould�mak�e�t�w�o�righ�t�x;angles�in�one�of�those�triangles.�E'Hence����Wno��Din��9terior�segmen�t�emanates��Cfrom��P�'�on�the�same�side�of��P�H�Q��as��A�.�kThen����Wtriangle����T�
��is�congruen��9t�to����AQR�>�.�"ZSince�not�b�A�oth�angle��C���QR��6�and�angle����W�B�r�R�>Q����can�b�A�e�righ��9t���angles�(else��QC�Tj�and��R�B�#7�w��9ould�b�A�e�parallel),�ĭan�in�terior����Wsegmen��9t�#�m�ust�#�emanate�from�one�of�them;�+$relab�A�eling,�'�w��9e�can�assume�it�is����W�R�>�.���The��other�endp�A�oin��9t�of��this�segmen�t�m�ust�b�A�e���C����.���The�other��in�terior����Wsegmen��9t��Zhas�one�endp�A�oin�t�at��P�H��,���and�its�other��Yendp�A�oin�t�m�ust�also�b�A�e�at��C����,����Wsince���there���is�no�in��9terior�v�ertex.��Then�b�y���the�lemma,����C���P��i�is�p�A�erp�endicular����Wto�4�QR�>�.�.Also�b��9y�3the�lemma,���P�H�Q��m�ust�3b�A�e�p�erp�endicular�3to��AC����.�.But�then����Wtriangle�<��C���QP����has�righ��9t�angles�<�at�b�A�oth��P��and��Q�,���con��9tradiction.���That����Wcompletes�Tthe�pro�A�of.��Bx���W�4��oL�Strict�ff7-tilings��阍���R�color push gray 0��Theorem���4�	color pop�������src:1015 SevenTriangles.tex�Ther��e�N<is�no�strict�7-tiling.����W�src:1017 SevenTriangles.texPr��o�of�.��Consider�Z�a�Z�strict�7-tiling.�Since�it�is�comp�A�osed�of�Z�7�triangles,�lthe����Wangles�|qmak��9e�a�total�of�|p7��R��.��zOf�that�7���,��there�is�����in�|pthe�corners�of�the�large����Wtriangle,�&�and�#62��u��for�#5eac��9h�in�terior�#6v�ertex,�&�and���u��for�#6eac�h�b�A�oundary�v�ertex����W(i.e.�B]v��9ertex�̣lying�on�an�edge�of�̢the�large�triangle�but�not�in�a�corner).����WTherefore�Jw��9e�Jha�v�e�either:���zero�Jin�terior�v�ertices�Jand�6�b�A�oundary�v��9ertices,����Wor�x
one�in��9terior�v�ertex�and�4�b�A�oundary�v�ertices,���or�t�w�o�in�terior�v�ertices�and����Wt��9w�o�Tb�A�oundary�v��9ertices.�pW��:�e�consider�these�three�cases�one�b�y�one.����d��src:1024 SevenTriangles.texCase��k1:��Zero�in��9terior�v�ertices�and�6�b�A�oundary�v�ertices.��#Since�there�are����W6��.b�A�oundary�v��9ertices,��dthere�are�9�sides�of��-triangles�on�the�b�oundary�and����W(21�SN��SO�9)�=�2���=�6���in��9terior�edges.��IIf�at�an�y�b�A�oundary�v�ertex,�ʑonly�one�in�terior����Wside��terminates�there,�ڴthen��
�O��m��9ust�b�A�e�a�righ�t�angle.�Assume�that��
�O��is�not����Wa�[�righ��9t�angle.��Then�consider�a�b�A�oundary�v�ertex��P����on�[�side��AB�r��,�mPthe�next����Wv��9ertex��to�ڿ�A�.�l�It�m�ust��connect�to�v��9ertex��Q��on�side��AC����,�the�next�v��9ertex����Wto�B �A��(else�no�side�can�escap�A�e�from��Q�).���The�other�edge�from��P���m��9ust�not����Wconnect���to�side����AC����,���or�else�no�edge�can�escap�A�e�from��Q�.��+So��P����connects����Wto�B�a�B�v��9ertex��R�S%�on�side��B�r�C����.��'The�second�edge�from��Q��m��9ust�connect�either����Wto�>��R�N��or�to�>�a�v��9ertex��S����on��B�r�C���b�A�et�w�een��C���and��R�>�.��TAssume�the�>�latter.�But����Wthen,��|a���second�edge���from��S�z��m��9ust�in�tersect�either����P�H�R���or��P�Q�,��{so�that���case����Wis��@imp�A�ossible�and��?�Q��connects�to��R�>�.�iConsider�another�b�A�oundary�v��9ertex��U����.����WRelab�A�eling,���w��9e��can��assume��U�}��lies�on��P�H�B����or��R�>B�r��.��XAssume�rst�that��U�}��is�on����W�R�>B�r��.��^The�M�other�M�end�of�b�A�oth�edges�lea��9ving��U�F��m�ust�M�b�A�e�at��P����or�at�another�����>�color push gray 0�����10��Y�	color pop�����?�y�������>�color push gray 0�Y�	color pop���?������W�v��9ertex�|��V���on��P�H�B�r��,��Zsince�there�are�no�in�terior�v�ertices.��Since�t�w�o�edges�lea�v�e����W�U����,���there��ais��bsuc��9h�a�v�ertex��V����on��P�H�B��and��U���V��is�an�edge.��uThen�either��U���P��D�or����W�R�>V�P�m��9ust�qb�A�e�qan�edge,���to�tile�quadrilateral��P�H�R�>U���V�8�.��Assume�it�is��R�>V�8�.��Then����Wthe���second�edge���lea��9ving��U��t�m�ust�terminate�at�another���b�A�oundary�v�ertex��W����W�on�5��V�8B�r��,�=�and�there�are�no�5�more�edges�to�lea��9v�e�5��W�H��,�so�the�angles�5�at��W�~g�are����Wrigh��9t�G9angles�G:and��
�h}�=�����R�=�2.�� Similarly�if��U���P���is�an�edge�instead�of��R�>V�8�,�S�the����Wsecond�edge�lea��9ving��V�I�m�ust�terminate�at�another�b�A�oundary�v�ertex��W�M��on����W�U���B�r��,��mand��4there�are��3no�more�edges�to�lea��9v�e��4�W�H��.��Hence�the�assumption�that����W�U����is���on��R�>B�*��has�led�to�a�con��9tradiction�(under�the�assumption�that��
�;Y�is����Wnot���a�righ��9t�angle).���The�remaining�alternativ�e���is�that��U����is�on��P�H�B�r��.���But����Wthen���the�argumen��9t�pro�A�ceeds�similarly:�תthe�other�end�of�b�oth�edges�lea��9ving����W�U��s�m��9ust�ϐb�A�e�Ϗat��R����or�at�another�v��9ertex��V����on��R�>B�r��.�K"Since�t�w�o�ϐedges�lea�v�e����W�U����,�Sthere�F�is�F�suc��9h�a�v�ertex��V�T��on��R�>B��e�and��U���V�T��is�F�an�edge.���F��:�rom�that�p�A�oin�t����Wthe�kargumen��9t�is�exactly�the�ksame�as�in�the�case��U�c��is�on��R�>B�r��.��Hence�the����Wassumption�0�that��
���is�a�righ��9t�angle�has�0�led�to�a�con�tradiction.�oGHence��
���is����Wa�Trigh��9t�angle.����d��src:1050 SevenTriangles.texSix�e�b�A�oundary�v��9ertices�means�that�9�edges�of�triangles�lie�on�the�b�ound-����Wary��-of��,the�large�triangle.�cSince�there�are�only�sev��9en�triangles,��hthat�means����Wthat�?"there�exist�?!t��9w�o�?"triangles�with�t��9w�o�edges�?"on�the�b�A�oundary��:�,�I�i.e.���there����Ware�5bt��9w�o�5atriangles�lo�A�cated�at�v��9ertices�of�triangle��AB�r�C��I�whic��9h�do�not�share����Wthat���v��9ertex�with�an�y�other���triangle.���Hence�the�nal�triangle��AB�r�C�K��is�simi-����Wlar�^dto�triangle��T�H��.��uSince�there�are�supp�A�osedly�sev��9en�^ccopies�of��T��G�tiling��AB�r�C����,����Wthe���similarit��9y�factor���is������?�p���XA���?�aH���u���7�����?.��BLet�us�supp�A�ose�that�the�nal�triangle��AB�r�C����W�has�
angle�����and��A�,�{angle����8�at��B�r��,�and�a�righ��9t�angle��
����at��C����.�!�Consider�the����Wtriangle�[_�T�����1����of�the�tiling�that�has�a�v��9ertex�at�[^�A�.��It�m�ust�ha�v�e�angle���]&�at����Wthat���v��9ertex.�vkLet��P�ы�b�A�e�its���v�ertex�on����AC�,��and��Q����its�v�ertex���on��AB�r��.�vjThen����W�P�H�Q�^�=��a�.���Let���T�����2�� ��b�A�e��the�triangle�that�shares�side��P�H�Q��with��T�����1��*��,�.Eand�let����W�R�m��b�A�e�]its�]�third�v��9ertex.���Since�there�are�no�in��9terior�v�ertices,�o��R�m��m�ust�lie�]�on����W�AB�r��,��Nor��on��AC����,�or�on��B�r�C����.�F�Case�1a,��R��Z�lies�on��AC����.�F�Then�b�A�oth��T�����1�����and����W�T�����2��� �ha��9v�e��xtheir��yrigh�t�angle��yat��P�H��.�A��R�ܷ�is�not�equal�to��C�pa�since�it�is�2�b��from����W�A�,�8�while�1��C�՛�is������?�p���
����?�aH���u���7������b�1��from��A�.�q�Consider�the�triangle��T�����3��\Z�that�shares�side��QR����W�with�2��T�����2��*��.�tLet��S����b�A�e�its�2�third�v��9ertex.�tWhat�is�angle��QR�>S���?�tIt�cannot�b�A�e��
����,����Wsince��side���c��of�triangle��QR�>S��is��QR��.��It�cannot��b�A�e���x,�,��zsince�that�w��9ould�mak�e����Wangle�T��AR�>S����equal�T�to���:d�+�8���x,�,�d�a�righ��9t�angle,�d�and�side��R�S����equal�T�to��a�,�d�whic��9h����Ww��9ould��lea�v�e��S����in�the�in�terior�of���AB�r�C����.�/Hence�angle��QR�>S����m��9ust�b�A�e�����,�and����Whence���angle��AR�>S�d��=���2���<����R�=�2,�[so��S�a��m��9ust�lie�on��AB�r��.�yLThat�mak�es�the����Wtotal�@<angle������at��Q��equal�to�3��x,�,�J�so���Qt�=��I��R�=�3,�J�����=���=�6,�J�so�@<�a��=���Hsin���N����=�1�=�2����Wand����b�KH�=��cos��8j��M�=���KG���?�p�������?�aH���u���3�������=�2.�jZAn��9y�linear�com�bination�of��a�,��u�b�,�and��c��is�th��9us�of����Wthe�a�form�a��p�A�+�A�q���R����?�p�������?�aH���u���3������;���but��AC����=�������?�p���
�G���?�aH���u���7������is�the�sum�of�sev��9eral�sides�of�the�basic����Wtriangle,�J�con��9tradiction.���This�@con�tradiction�@disp�A�oses�of�Case�1a.���Case�1b,����W�R��G�lies��
on��	�AB�r��.�U�Then�the�righ��9t�angles�of��T�����1�����and��T�����2���are�b�A�oth�at��
�Q�.�U�Let����W�T�����3��&.�b�A�e���the���triangle�sharing�side��P�H�R���with��T�����2��*��,��and�let��S�~��b�A�e�its�third�v��9ertex.����WIf��[�S�7`�lies��\on��AC����,��then�angle��R�>P�H�S��m��9ust�b�A�e��\either���,��or�����;��if�it�is����"�then����W2��-��+��b��5t�=�3���R��,�N�whic��9h�is�imp�A�ossible�since����(�+��a�����=���R�=�2�and�����<��R��.��If�it����Wis��l����then�3����=�h���R��,���whic��9h�is��kimp�A�ossible�as�in�the�previous�case.���Hence��S����W�do�A�es�8Vnot�lie�on��AC����.��u�S��[�cannot�lie�on��AB���as�8Uthat�w��9ould�mak�e�angle��P�H�R�>S����W�more�1�than���R�=�2.�r4No��9w��T�����1��*��,�9�T�����2���,�9and�1��T�����3��\��all�1�share�v�ertex��P�H��,�9but�1�since��S����do�A�es����Wnot�&(lie�on��AC����,�*]a�fourth�triangle�&)�T�����4��P��shares�that�v��9ertex�to�A�o,�and�has�angle����Wat���least������there.��kIf��T�����3��Ƥ�has�angle���(�at��P����then�the�total�angle�at��P��is�at����Wleast�k�3�����+�G���#W>�!��2���+�2��#W�=�!���R��,��+con��9tradiction.�BHence�k��T�����3���@�has�angle���ma�at��P�H��,�����>�color push gray 0�����11��Y�	color pop�����^�y�������>�color push gray 0�Y�	color pop���?������W�and�Ƶ�P�H�QR�>S�I��is�a�paralleogram.�;Let�triangle�ƴ�T�����4���\�b�A�e�the�triangle�sharing�side����W�P�H�S�,��=����b�#��with�#��T�����3��*��.�G�The�angle�of��T�����4��Ng�at��P�l��m��9ust�b�A�e�either���%��or�a�righ��9t�angle,����Wbut�^a�righ��9t�angle�is�to�A�o�large,�p>since�the�other�angles�at��P����total���R�=�2�>�+�>���x,�.����WThe���righ��9t���angle�of��T�����4���,�m��9ust�therefore�b�A�e�at��S���,��Hmaking��S�&��an�in��9terior�v�ertex,����Wcon��9tradiction.�޹That�\.disp�A�oses�of�Case�\-1b,��7�R�ll�lies�on��AB�r��.�޹But�since�the�righ��9t����Wangle���of����T�����1��Đ�m��9ust�lie�at�either��P����or��Q�,���Case�1a�and�Case�1b�are�exhaustiv��9e,����Wso�TCase�1�has�b�A�een�sho��9wn�to�b�e�imp�ossible.����d��src:1080 SevenTriangles.texCase��w2:��one�in��9terior��xv�ertex,���and��wfour�b�A�oundary�v��9ertices.���Then�there����Ware�6.(21������7)�=�2�s0=�s17�6-in��9terior�edges.�~�There�can�b�A�e�at�most�one�triangle����W�T�����i��7k�that�
�has�
�one�v��9ertex�on�eac��9h�side�of��AB�r�C����.��^(Call�suc��9h�a�triangle�an����W\in��9terior�rttriangle".)�3�Hence�rusix�or�sev�en�triangles�ruha�v�e�one�or�rumore�sides����Won�d�the�b�A�oundary�of�d��AB�r�C����.�
With�four�b�oundary�v��9ertices,�x\there�are�sev�en����Wb�A�oundary���segmen��9ts�to���b�e�accoun��9ted�for.��If�there�is���an�in�terior�triangle,����Wthen�n�one�of�the�remaining�six�m��9ust�accoun�t�for�t�w�o�b�A�oundary�segmen�ts,����Wso��one��
of�the�v��9ertices��A�,����B�r��,���or��C�Z��is�not�\split",���i.e.��shared�b��9y�t�w�o��
or�more����Wtriangle���T�����i��,r�.��If��there�is�no�in��9terior�triangle,���then�eac��9h�of�the�sev��9en�triangles����Wm��9ust�
Zaccoun�t�
[for�exactly�one�b�A�oundary�segmen��9t,��whic�h�
[means�that��al�x�l��of����Wthe�Tv��9ertices��A�,��B�r��,�and��C��<�are�\split".����d��src:1089 SevenTriangles.texCase��j2a:�{There�is�an�in��9terior�triangle.�"Let��T�����1��$�b�A�e�that�triangle,���ha�ving����Wv��9ertex����P���on��AB�r��,��v�ertex��Q��on��B�r�C����,��and�v�ertex��R���on��AC����.�.That�creates����Wthree���triangles��B�r�P�H�Q�,��]�AP�R�>�,��\and����QR�C����.���The���single�in��9terior�v�ertex�m�ust����Wo�A�ccur�<�in�<�the�in��9terior�of�one�of�these�triangles�(since�this�is�a�strict�tiling,����Wit�9Ecannot�o�A�ccur�on�the�b�oundary�of��T�����1��*��).��BRelab�eling�the�v��9ertices�if�neces-����Wsary�4!w��9e�can�assume�that�it�o�A�ccurs�in�4"triangle��AP�H�R�>�.�x�A�t�least�three�edges����Wlea��9v�e�~that�~in�terior�v�ertex,��Hso�triangle�~�AP�H�R��U�is�divided�in��9to�at�least�three����Wtriangles���congruen��9t�to��T�H��.��In�fact�it�m�ust�b�A�e�divided�in�to�exactly�three����Wtriangles,�W'since�I�at�least�I�three�are�needed�for��T�����1��*��,��B�r�P�H�Q�,�W&and��QR�>C����,�so�the����Wp�A�ossibilities��are�three��or�four,��but�there�is�no�4-tiling�with�an�in��9terior�v�er-����Wtex,���b��9y��^Theorem�3�Hence��AP�H�R����is��]3-tiled,���and�there��^is�only�one�3-tiling����Wwith�{�an�{�in��9terior�v�ertex,��}b�y�{�Theorem�2.�PTherefore��T�Ŀ�is�the�triangle�with����W��c��=�a�����=���R�=�6��%and��$�
��Y�=�a�2��=�3,��Xand�triangle��AP�H�R��b�is��%equilateral.���Consider����Wthe���angles�at��R�>�:��angle��AR�P����is�����R�=�3,��and��P�H�R�Q��is���R�=�6,��so��QR��/�is�p�A�erp�endic-����Wular�||to�|}�AC� d�and�therefore�angle��QR�>C� d�m��9ust�b�A�e�comp�osed�||of�three�angles����W����.��That��means��that�triangle��QR�>C�`��con��9tains�three�smaller�triangles,�θwhic��9h����Wmak��9es�?sev�en�coun�ting�the�three�>in��AP�H�R�}�and��T�����1��*��,�:�lea�ving�none�to�co�v�er����W�B�r�P�H�Q�.�pThis�Tdisp�A�oses�of�case�2a.����d��src:1102 SevenTriangles.texCase�iw2b:�ĵNo�ivin��9terior�triangle,�~and�all�v��9ertices��A�,�~�B�r��,�and�iw�C�
^�are�split.����WThen��rthere��qis�an�in��9terior�edge�emanating�from�eac��9h�of��A�,��9�B�r��,��8and��C����.��An��9y����Wpair��"of��!these�m��9ust�in�tersect�in��!an�in�terior�v�ertex.���But�there�is��!only�one����Win��9terior���v�ertex,�7�so�they���all�in�tersect�in���a�common�p�A�oin��9t��P�H��,�7�the�in�terior����Wv��9ertex,��<forming���three�triangles��AB�r�P�H��,��=�AC���P��,�and����B�r�C���P��.��These�triangles����Ware��4tiled�b��9y��T�H��,��lwithout�in�terior��5v�ertices,��ksince�there��4is�only�one�in�terior����Wv��9ertex.�3Hence��none��of�them�is�3-tiled.�3A��4-tiling�requires�a�b�A�oundary����Wv��9ertex��Son�eac�h��Tside,��Swhic�h�w�ould�mean�another��Tin�terior�v�ertex,��Sso�none����Wof�Ǖthem�ǔis�4-tiled�either.�32They�cannot�all�b�A�e�2-tiled,��$as�that�w��9ould�use����Wonly�wesix�triangles.�B�That�lea��9v�es�weonly�the�p�A�ossibilit��9y�that�t�w�o�of�them�are����Wcongruen��9t��to���T���and�the�other�is�5-tiled;�%�relab�A�eling�v��9ertices�if�necessary��:�,����Ww��9e���can���assume�it�is��AB�r�P�ˋ�that�is�5-tiled.�dnSince�there�are�four�b�A�oundary����Wv��9ertices��Tthey��Sm�ust�all�b�A�e�on��S�AB�p	�and�the�v��9e�triangles�all�share�v��9ertex��P�H��,����Wwith��one��side�con��9tained�in��AB�r��.��This�is�imp�A�ossible�for�sev��9eral�reasons,���for�����>�color push gray 0�����12��Y�	color pop����
���y�������>�color push gray 0�Y�	color pop���?������W�example,�$�since��just��t��9w�o�triangles�share��eac�h�b�A�oundary��v�ertex,�$�all�those����Wangles��m��9ust�b�A�e��righ�t�angles,�*con�tradicting�the�fact�that��all�those�edges����Wmeet�Tat��P�H��.�pThis�disp�A�oses�of�Case�2b,�and�hence�of�Case�2.����d��src:1113 SevenTriangles.texCase�z3:���t��9w�o�zin�terior�v�ertices�and�zt�w�o�b�A�oundary�v�ertices.�J�Then�there����Ware�n�(21������5)�=�2���=�8�in��9terior�edges.���Supp�A�ose�n�rst�that�t�w�o�n�b�A�oundary�v�ertices����W�P��M�and�Ui�Q�Uj�o�A�ccur�on��AB�r��,�eowith��P��M�adjacen��9t�to��A��and��Q��adjacen��9t�to��B�r��.�ܱLet����W�U���and����V���b�A�e�the�in��9terior�v�ertices.�DIf��
�n\�is�not���a�righ�t�angle,��Pthen�t�w�o�edges����Wm��9ust���lea�v�e��P�E��and�t�w�o�edges�m�ust�lea�v�e��Q�.�EOne�of�the�edges�from��P�H��,��and����Wone��_from��`�Q�,�*can�go�to�an�in��9terior�v�ertex,�*and�one��`to�v�ertex��`�C����.�tOne�more����Wedge�� can��!go�from��P�(�or�from��Q��to�an�in��9terior�v�ertex,��but�after�that��!w�e�are����Wblo�A�c��9k�ed{there���is�no�place�to���put�the�rest�of�the�8�edges.��2Hence�the�t��9w�o����Wb�A�oundary���v��9ertices���do�not�o�A�ccur�on�the�same�side�of�the�large�triangle.��Sa��9y����W�P��e�o�A�ccurs���on����AB�7�and��Q��o�ccurs�on��B�r�C����.���Then�eac��9h�of��P��e�and����Q��can�connect����Wto�T*b�A�oth�T)in��9terior�v�ertices,���and�one�T)of�them�can�connect�to�an�opp�A�osite����Wv��9ertex��Qof��Rthe�large�triangle,��but�that�is�not�enough�edges.�iTherefore��P����W�and����Q��do���not�connect�to��C����.�T��:�o�use�up�8�edges,��Hw��9e�m�ust���ha�v�e�an�edge����Wconnecting���U����and���V�8�,� pand�one�of��U����and��V��,� psa��9y��V��,� pconnects��to�b�A�oth��B����W�and���C����,��while��U����connects�to��A�.��But��then,��there�are��exactly�four��angles�at����W�V�8�,���totaling���2��R��.��WThat�means�t��9w�o���of���them�m��9ust�add�to�at�least���R��,���whic��9h����Wmeans�T�
����is�a�righ��9t�angle.����d��src:1123 SevenTriangles.texIf��Jjust��Kthree�edges�emanate�from��V����then�there�are�exactly�three�angles����Wat�x��V�8�.��-If�three�angles�add�to�2��R��,���they�m��9ust�x�all�b�A�e��
����,�since�2�
�X2�+�Ԥ��	�<����2��R��.��-But����Wthen�u|�
�T�=���2��R�=�3,��tcon��9tradicting�our�u{conclusion�that��
��
�is�a�righ��9t�angle.��(Hence����Wat���least�four�edge�emanate�from��U����and�four�from��V�8�,��for�the�required�total����Wof�M�8.��Ev��9ery�M�one�of�the�eigh��9t�in�terior�edges�M�then�has�one�end�at��U�F��or�one����Wend��at��V�8�.��The�four��edges�emanating�from��U����go�to�the�edge�v��9ertices��P����and����W�Q�y��and�y�to�t��9w�o�y�v�ertices�of�y�the�large�triangle,���sa�y��A�y��and��B�r��.���V����m�ust�lie�y�in�one����Wof��the��four�regions�formed�b��9y�the�angles�at��U����;�քthree�of�those�are�triangles,����Wwhic��9h��slea�v�e�only�ro�A�om��rfor�three�edges�to�emanate�from��V�8�.��Hence��V�߫�m�ust����Wlie��in��the�quadrilateral��U���P�H�C���Q�,��Band�m��9ust�connect�to�all�four�corners�of����Wthat���quadrilateral.���No��9w�w�e�ha�v�e�v�e�edges�emanating���from��V����and�four����Wfrom���U����,�!+and�as�ab�A�o��9v�e,�all��the�angles�at��U���are�righ��9t�angles.�8�No�w�consider����Wthe��angles�at���P�H��.���Angle��AP�U����and�angle��U���P�V��are��either�����or���x,�,�>2since����Wthose��ntriangles�ha��9v�e��mtheir��nrigh�t�angle�at��m�U����.���Hence�angle��V�8P�H�B�#�is��
����,��6a�righ��9t����Wangle.�#But��kthen��l�P�H�C�5��=����c�,�gsince�it�is�opp�A�osite�the�righ��9t�angle�at��V�8�,�gand�on����Wthe�;�other�;�hand�it�is�less�than��c�,�Esince�it�is�opp�A�osite�angle��V�8B�r�P�H��,�Ewhic��9h�is����Wnot�Ta�righ��9t�angle.�pThis�con�tradiction�eliminates�Case�3.����d��src:1134 SevenTriangles.texThat�Tcompletes�the�pro�A�of�of�the�theorem.��Bx���W�5��oL�Non-strict�ff�X�Qffcmr12�7�-tilings��阍���R�color push gray 0��Lemma���2�	color pop������src:1140 SevenTriangles.tex�Supp��ose���that�a����7�-tiling�c�ontains�a�non-strict���vertex�of�typ�e����W�3�5:�1�.�N,Then��Jthe��Itile�is�a�right�triangle,����c�5�=�3�a�,�and��Jthe�smal�x�lest��Iangle������W�satises�N<the�e��quation���sin��4B�����=���1�=�3�.����W�src:1145 SevenTriangles.texPr��o�of�.��kLet�Q�the�maximal�Q�segmen��9t�b�A�e��P�H�Q�,�`�running�from��P����in�the�\north"����Wto���Q���in�the�\south."��Supp�A�ose�three�triangles��T�����1��*��,�	��T�����2���,�	�and��T�����3��1v�o�A�ccur�on��the����Ww��9est���side���of��P�H�Q�,���meeting��P�Q����at�v��9ertices��U����and��V�8�,���and��T�����4���G�lies�on�the�east����Wside���of��P�H�Q�.��Let��c����b�A�e�the�longest�side�of��T�H��;��nthen��c��r�=��q3�a��or��c��r�=��q2�a�t|�+�t}�b�.����WIt��is��imp�A�ossible�that�the�three�congruen��9t�triangles�ha��9v�e��a�common�v��9ertex�����>�color push gray 0�����13��Y�	color pop����\�y�������>�color push gray 0�Y�	color pop���?������W�S���,��Yso�ÿthat��S�P�H�U����,��Y�S�U�V�8�,��Yand�ÿ�S�V�Q�ÿ�are�þcongruen��9t�triangles.�'�Hence�there����Ware��t��9w�o�distinct�p�A�oin�ts��S�u��and��R���suc�h�that���T�����1��-�=�h�S��P�H�U��f�and��T�����3���=�h�R�>V�8Q����W�are�4Jt��9w�o�4Iof�the�triangles�in�the�tiling.�yP�T�����2��^��ma��9y�ha�v�e�4Ia�side��S��U�--�in�common����Wwith���T�����1��=�or�a�side��R�>V����in�common��with��T�����3��*��.��6In�either�case�the�common����Wside���is�p�A�erp�endicular���to��P�H�Q��and��T�8��is�a�righ��9t�triangle,�&�so��
����=����R�=�2�and����W��_q�+�]���Џ�=�Xd��R�=�2.���W��:�e��ha��9v�e��P�H�U�QH�=�Xc�U���V�f��=��a�,��Jand��S��V�f��=�Xd�c��=�3�a��or�2�a�]��+�]��b�.���If����W�S��V���=���3�a�.��then��.�sin��������=��U���V�=S�V���=�1�=�3�.�as�claimed�.�in�the�lemma.�hThe�case����W�c����=�2�a���+��b��<�is��;imp�A�ossible,��
since�no�righ��9t�triangle�has�sides��a�,��b�,�and��<2�a���+��b�.����d��src:1158 SevenTriangles.texIf,�Z�on�L�the�L�other�hand,�Z��T�����2��wc�do�A�es�not�ha��9v�e�L�a�side�in�common�with��T�����1��wc�or����W�T�����3�����then��9there��:will�b�A�e�altogether�v��9e�triangles�on�the�left�of��P�H�Q��sharing����Wv��9ertices�Ɵ�U����and��V�8�.�3Let��W���b�A�e�ƞthe�w�est�v�ertex�ƞof��T�����2��*��.�3Then��W���cannot�lie�on����W�S��P�H��,���since���if�it�do�A�es,��W�H�U����is�longer�than�b�A�oth����S��P����and��S�U����,���but�one�of�those����Wsides�M�m��9ust�b�A�e�the�longest�side�of�triangle��T�H��,�[�since�the�third�side�of��S��P�U����W�is���less�than��P�H�Q�,���whic��9h�is�one�side���of�the�cop�y�of��T����on�the�righ�t�of��P�H�Q�.����WSimilarly��:�,�D�W�`��do�A�es��not��lie�on��R�>Q�.�#}Let��T�����5��BU�and��T�����6���b�A�e��the�other�copies�of��T����W�in�uT�M�n7�sharing�v��9ertices��U�n8�and�uS�V�8�,��Tresp�A�ectiv�ely��:�.�<oLet��M�n8�b�A�e�uSthis�six-triangle����Wconguration.���W��:�e��}claim��~that�the�b�A�oundary�of��M��a�con��9tains�at�least�v��9e����Wnon-straigh��9t��
angles.�WA�t��R�G�there�is�either��	another�non-strict�v�ertex��	with�a����Wnon-straigh��9t�Bangle,�Eor�at�least�A(if��R� ��is�a�v��9ertex�of��T�����2��*��)�the�b�A�oundary�of��M����W�is�3not�2straigh��9t.�.
Similarly�at��S���.�.There�is�an�angle�at�the�east�v��9ertex�of��T�����4�����W�(the���triangle�on���the�righ��9t�of��P�H�Q�).��W��:�e�claim�there�are�also�non-straigh��9t����Wangles���at��P���and��Q�.��IF��:�or���those�to�b�A�e�straigh��9t�angles,���w�e���w�ould�ha�v�e���to�ha��9v�e����Wthe�isum�of�t��9w�o�iangles�of��T�KK�equal�to���R��.�"But�at�(one�of��q)��P�KL�or�h�Q�,�2the�angle����Wof����T�����4����is�the�small�angle�����,��Ethe�angle�from��T�����1���or��T�����3���is�not�����,��Esince���ȩ�is����Wthe�a�angle�a�at��S���or��R�>�;���so�the�sum�of�the�t��9w�o�a�angles�is�less�than����+�(in�fact����Wit�(.is���mt�����x,�).�T�A��9t�the�other�of��P�q�or�(/�Q��w�e�w�ould�need����Z�to�(/b�A�e�a�righ�t�angle����Wto�O"create�O!a�straigh��9t�angle.���In�that�case��T�����2��y��w��9ould�ha�v�e�O"a�side�in�common����Wwith���T�����1�����or��with��T�����3��*��,��%con��9tradiction.��Hence�there�are�v�ertices�of���M�|��(at�least)����Wat�n��P�H��,��M�Q�,��L�R�>�,��S���,�and�the�n�third�v��9ertex�of��T�����4��*��{v�e�in�total.�)+Supp�A�ose�it�w�ere����Wp�A�ossible�'�to�place�one�more�cop��9y�of��T�pk�next�'�to��M� l�so�as�to�form�a�triangle.����WIf�$the�%triangle�is�placed�to�the�righ��9t�of��P�H�Q��against�one�of�the�sides�of��T�����4��*��,����Wthat��Dma��9y��Celiminate�a�v�ertex��Cat��P��'�or��Q�,���but�will�lea��9v�e�a��Dv�ertex�at��Cthe�other����Wof��P�e��or��Q�,�as�w��9ell�as�creating�one�more�new�v�ertex�and�lea�ving�three�old����Wones{to�A�o�'3man��9y�v�ertices�for�a�triangle.�RIf�the�triangle�is�placed�to�the�left����Wof�>F�P�H�Q�,�H�against��T�����1��h��or��T�����3��*��,�H�again�>Eit�ma��9y�eliminate�a�v��9ertex�at��P��)�or��Q�,�H�and����Wp�A�ossibly��Yat��S�C_�or��R�>�,��Xbut�it�will��Zcreate�a�new�v��9ertex�and�lea�v�e��Zat�least�three����Wold�Aones.���Placing�Ait�an��9ywhere�else�will�lea��9v�e�v�ertices�Aat�all�three�v��9ertices����Wof��M�T�����4��*��;��but�since��Lpart�of��M��1�exists�outside�the�con��9v�ex�h�ull��Mof�those�v��9ertices,����Wthose�~�cannot�b�A�e�~�the�v��9ertices�of�a�triangle�con��9taining��M����.�YfHence,��ea�tiling����Wwith�z�a�z�v��9ertex�of�t�yp�A�e�z�3�;7:�1�in�z�whic��9h��T�����2�����do�A�es�not�ha��9v�e�z�a�side�in�common����Wwith����T�����1��(8�or��T�����3���cannot�o�A�ccur�in�a�7-tiling.��That�completes�the�pro�of�of�the����Wlemma.������R�color push gray 0��Lemma���3�	color pop������src:1185 SevenTriangles.tex�A�N<�7�-tiling�c��annot�c�ontain�a�maximal�se�gment�of�typ�e��3���:�2�.����W�src:1188 SevenTriangles.texPr��o�of�.�=Consider��a�non-strict��tiling�con��9taining�a�maximal�segmen��9t�of�t�yp�A�e����W3��}:��|2.���Let�G��T�����1��*��,��\�T�����2���,��]and�G��T�����3��ri�o�A�ccur�on�the�left�of�the�(v��9ertical)�maximal����Wsegmen��9t����P�H�Q�,��\with���v�ertices��U��r�and��V����on��P�H�Q�.��Already�v�e���triangles�will����Wha��9v�e�}nv�ertices�}oon�the�maximal�segmen��9t��P�H�Q�.�T�If�they�do�not�ha��9v�e�}osides�in����Wcommon�k�then�k�three�more�triangles�will�b�A�e�required�to�ll�in�the�gaps{more����Wthan�d�sev��9en�d�altogether.�
�Sa�y�then�d�that��T�����1���`�and��T�����2���ha��9v�e�d�a�side�in�common.�����>�color push gray 0�����14��Y�	color pop����$�y�������>�color push gray 0�Y�	color pop���?������W�Let���U���b�A�e�the�v��9ertex��that��T�����1��9��and��T�����2��9��share�on��P�H�Q�.�OBy�Lemma�1,�9�
�T�=�����R�=�2����Wand�2�T�����1��,��and�1�T�����2���b�A�oth�ha��9v�e�a�1righ�t�angle��
����at�1�R�>�.��
Supp�A�ose,�=hfor�pro�of�b��9y����Wcon��9tradiction,��Kthat��the�sides�of��T�����1����and��T�����2���on��P�H�Q��are�equal�to��a�,��Kand�that����W�T�����1��47�and�	��T�����2��46�share�a�common�side�	�(but�not�necessarily�a�common�v��9ertex����Ww��9est��Lof��P�H�Q�).�>YBut�then�they��M�do��share�a�common�v�ertex��S�NR�w�est�of��P�H�Q�,����Wsince��they��ha��9v�e�their��angles�at��U����equal�(b�A�oth�righ��9t�angles)�and�their�w��9est����Wangles�
�b�A�oth�equal�to�
�����,��hence�the�angles��S��P�H�U���and��S�V�8U���are�b�A�oth���x,�,��and����Wthe��sides��opp�A�osite�are�b�oth��b�,�Vso�the�common�v��9ertex���S����is��b��a�w�a�y��from��U����.����WThen�triangle��S��P�H�U���is�congruen��9t�to��S�V�8U����.�Let��T�����4��/��and��T�����5���b�A�e�the�triangles����Won���the���east�side�of��P�H�Q�,���sharing�v��9ertex��R����on��P�H�Q�,���and�let��T�����6���O�b�A�e�another����Wtriangle�Jw��9est�Iof��P�H�Q��sharing�v�ertex�I�V����(there�m�ust�Ib�A�e�one�since��T�����2�����do�es�not����Wha��9v�e�qa�righ��9t�angle�qthere.)�/�Let��E��N�b�A�e�the�east�v��9ertex�of��T�����4��*��.�/�If��T�����4�����and��T�����5���,����Wthe�%t��9w�o�&triangles�on�the�east�of��P�H�Q�,��do�not�share�a�side�(and�hence�ha��9v�e����Wrigh��9t�ɷangles�ɸat��R�>�),���then�a�sev�en�th�ɸtriangle�m�ust�ɸo�A�ccur�b�et��9w�een�ɸthem,���and����Ww��9e��?ha�v�e�to�A�o��@man�y�v�ertices:����P�H��,��C�Q�,��D�W��,�and��@�S�MD�at��?least.�iHence��T�����4�����and��T�����5�����do����Wshare�lXa�side,��and�their�angles�at��P��;�and��Q��are�acute,�and�their�angles�at����W�R���are��^righ��9t��]angles.��Hence��E��2P�;@�and��E�Q��are��^equal�to��c��and��P�H�Q��is�2�b�,��\since����Wit�Tcannot�b�A�e�2�a��as�it�is�the�sum�of�three�sides�of��T�����1��*��,��T�����2���,�and��T�����3���.����d��src:1209 SevenTriangles.texThen��othis��psix-triangle�conguration��M��T�has�v��9ertices�at��P�H��,��7�E��2�,��Q�,��S�?t�(the����Wshared�#/w��9est�v�ertex�#.of��T�����1��M��and��T�����2��*��),�&�and�the�w��9est�v�ertex��W�l�of��T�����3��*��.�FT��:�riangle����W�T�����6��wW�m��9ust�L�then�L�ll�the�angle�at��V�8�,�Z�or�else�the�sev�en�th�L�triangle�w�ould�need����Wto���touc��9h��V�8�,���lea�ving�more�than�three�exterior�v�ertices,���namely��P�H��,����Q�,��E��2�,����Wand�Tat�least�one�v��9ertex�w�est�of��P�H�Q�.����d��src:1214 SevenTriangles.texSupp�A�ose,�'�for���pro�of�b��9y�con�tradiction,�'�that��W�9��lies�on�line��S��Q�.���Then����Wangles�EK�V�8W�H�Q��and��V�W�H�S��P�are�EJrigh��9t�angles,�QIso��V�Q�,�QHthe�side�opp�A�osite�angle����W�V�8W�H�Q���in��triangle��T�����3��*��,��equals��c�,�and�triangle���S��V�8W�]��is�congruen��9t�to��QV�8W�H��.����W�M�`[�then�gvforms�gwa�quadrilateral,�{�and�along��P�H�Q��w��9e�see�2�a�D��+��c���=�2�b�.��Angle����W�W�H�S��V���=��z����,�O�b��9y�C�the�congruence�of�triangles��W�S��V�R+�and��W�QV�8�.��LThe�angles����Wof�
�triangle��S��P�H�Q��are������at��P��,�P����at��Q�,�and�hence��
�T�=�����R�=�2�at�
��S���.��This�angle����Wat�Y�S���is�also�equal�to�X�3����,�i�since�eac��9h�of�triangles��T�����1��*��,�i��T�����2���,�i�and�Y�T�����6�����has�angle����W��t��there.�6Hence�s6��0�=�.>��R�=�6,���so�s5�a��=�����P��aq�1��aq��&�fe�����2�����	?K�,��b��=������?�p���
㐟��?�aH���u���3�������=�2,���and�s6�c��=�.=1.�6But�s5then�the����Wequation�P�2�a�5��+��c����=�2�b��do�A�es�not�hold,�_�con��9tradiction.���Hence��W����is�do�es�not����Wlie�Ton�line��S��Q�.����d��src:1224 SevenTriangles.texHence���there�are�t��9w�o���v�ertices�of��M����w�est���of��P�H�Q��(either��S���and��W�E��or����Wv��9ertices�*of�*�T�����6��*��).�Z��M�"��th�us�has�*at�least�v��9e�v�ertices.�Z�T��:�o�reduce�*this�to�three����Wv��9ertices�0=b�y�0<placing�one�more�triangle�east�of��P�H�Q��is�imp�A�ossible.�m*Similarly����Wit���is���imp�A�ossible�to�reduce�the�n��9um�b�A�er���of�v�ertices���to�three�b��9y�placing�a�new����Wtriangle�u�along��P�H�S����or��W�Q�.�>DBut�then,��no�matter�where�else�w��9e�place��T�����7��*��,����W�P�H��,��y�E��2�,��xand����Q��will��remain�v��9ertices,�and��there�m��9ust�b�A�e�a�fourth�v��9ertex�w�est����Wof�<I�P�H�Q�,�Fso�the�result�cannot�b�A�e�a�triangle.��OThis�con��9tradiction�sho�ws�that����Ww��9e�Tcannot�ha�v�e��V�8U����=����P�H�U��=��a�T�as�w��9e�assumed�ab�A�o�v�e.����d��src:1230 SevenTriangles.texNo��9w�+w�e�drop�the�con�tradictory�assumption��V�8U����=����P�H�U��=��a�+�and�b�A�egin����Wanew.�f�Again��[w��9e�ha�v�e��Zrigh�t�angles�in��T�����1����and��T�����2����at��U����,���but�this��Ztime�one����Wof�ٛthem�ٜhas�side��b��along��P�H�Q��(not��c�,��since�that�m��9ust�b�A�e�opp�osite�ٜthe�righ��9t����Wangle���at����U����).���The�other�one�has�side��a��along��P�H�Q�,��`since�2�b�k,���a�e-�+�e.�b�>�c�.����WHence�ćthey�do�not�Ĉshare�a�common�w��9est�v�ertex.��Let��S�G��b�A�e�the�w�est�v�ertex����Wof�l3�T�����1�����and��X� ��the�w��9est�v�ertex�l4of��T�����2��*��.�!Again��T�����6�����will�ha��9v�e�l3to�b�A�e�placed����Wb�A�et��9w�een����T�����2���6�and��T�����3���with�a�v��9ertex�at����V�8�.��!That�will�giv�e�us�a�six-triangle����Wconguration��O�M��3�with�v��9ertices��P�H��,��
�E��2�,��Q�,���S���,��X��[�,�and��O�W�2�(the�w�est�v�ertex����Wof����T�����3��*��).��W��:�e�only�ha��9v�e���to���sho�w�that�placing�one�more�triangle��T�����7�����cannot�����>�color push gray 0�����15��Y�	color pop����;\�y�������>�color push gray 0�Y�	color pop���?������W�p�A�ossibly�{npro�duce�a�triangular�{mconguration.�N�(That�is�not��prima��Ofacie����W�imp�A�ossible���just���b�ecause�there�are�six���v��9ertices{it�could�happ�en�if����M����had����Wt��9w�o�y�collinear�sides�separated�b��9y�t�w�o�y�sides�forming�a�\notc�h"�in�to�whic�h��T�����7�����W�w��9ould��just��t{so�some�further�argumen��9t�is�required.)�8If�the�t��9w�o��triangles����W�T�����4��p��and�F8�T�����5���east�of�F9�P�H�Q��do�not�share�a�side,��qthen��T�����7��p��w��9ould�ha�v�e�F8to�b�A�e����Wplaced��meast�of��n�P�H�Q��b�A�et��9w�een��mthe�t��9w�o,���lea�ving��ma�v��9ertex�at��P��Q�(since��T�����1����has����Wan�J�acute�angle�J�at��P�H��),�XMas�w��9ell�as�v�ertices�at�J��X��C�and��S���,�XLwhic�h�are�distinct,����Wand���of���course�at�least�one�v��9ertex�east�of��P�H�Q�,��totaling�more�than�three.����WHence����T�����4��*>�and��T�����5���do�share���a�side;�t�hence�their�righ��9t�angles�are�b�A�oth�on����W�P�H�Q�kG�at�kFv��9ertex��R�>�.�GHence�their�angles�at��P��*�and��Q��are�acute,���and�triangle����W�P�H�E��2R����is��congruen��9t��to�triangle��QE�R�>�.��GHence��the�t��9w�o��sides��P�H�R����and��R�Q����W�are�A�equal,�M)and�A�either�equal�to��a��or�to��b�.��pThe�length�of��P�H�Q��is�th��9us�either����W2�a�D��or�2�b�D��(measured�from�the�righ��9t�side),�P|and�also�either�2�a�-��+��b�D��or�2�b�-��+��a����W�(measured�s.from�the�s-left�side).�5�Three�of�the�four�p�A�ossible�equations�here����Ware��immediately�imp�A�ossible,���lea��9ving�only��the�p�ossibilit��9y�2�b����=�2�a���+��b�;��}hence����W�b��l�=��k2�a�.�}No��9w�5�if�the�angle�of��T�����1��`,�at��P�~g�is�����,�=�then�the�5�south�w�est�v�ertex��S����of����W�T�����1��Ӽ�lies��on�the�north�side�of��T�����2��*��,��and��S�,�is�th��9us�not�on�the�con�v�ex�h�ull�of����W�M����,���and��rtriangle��T�����7���cannot�ll�the��sobtuse�angle������u��i��exterior�to��M��W�at��S���.����WHence���the�angle�of��T�����1����at��P�/��is�not�����;��gso�it�m��9ust�b�A�e���x,�.��Then�the�north�w�est����Wv��9ertex����X�;��of��T�����2���=�lies�on���the�south�side�of��T�����1��*��,��&and�th��9us�not�on�the�con��9v�ex����Wh��9ull�^�of��M����,�qbut�there�^�is�an�obtuse�exterior�angle�������?�����at��X��that�cannot����Wb�A�e�N�lled�b��9y��T�����7��*��.��\W��:�e�conclude�that�N�it�is�not�p�ossible�to�place��T�����7��y��to�create����Wa�Ttriangle.�pThat�completes�the�pro�A�of�of�the�lemma.������R�color push gray 0��Lemma���4�	color pop������src:1293 SevenTriangles.tex�A�N<�7�-tiling�c��annot�c�ontain�a�maximal�se�gment�of�typ�e��4���:�1�.����W�src:1296 SevenTriangles.texPr��o�of�.��Let����P�H�Q��b�A�e�a�(north-south)�maximal���segmen��9t�with�four�triangles����Won��the�left��and�one�on�the�righ��9t.��oOf�the�four�triangles�on�the�left,�D>w��9e����Wcannot��ha��9v�e�three�sharing�a�v�ertex��not�on��P�H�Q�,�<�so�the�minim�um�\tile"����W(consisting�b�of�all�b�the�triangles�touc��9hing��P�H�Q�)�con�tains�at�b�least�six�triangles,����Wand�Ԃcon��9tains�sev�en�without�making�a�triangle,�Nunless�the�four�triangles����Won�?the�>left�o�A�ccur�in�t��9w�o�pairs,�ٹeac�h�pair�?ha�ving�a�?common�side.�Z0Let����Wthe�y�northernmost�y�of�these�pairs�b�A�e��T�����2���u�and��T�����3��*��,���and�let��T�����4���u�and��T�����5���t�b�A�e�the����Wsouthern�Xpair.��Let��V�f<�b�A�e�the�Xv��9ertex�on��P�H�Q��shared�b��9y��T�����3�����and��T�����4��*��.��Let��T�����1�����W�b�A�e��Rthe�triangle�on�the��Srigh��9t�of��P�H�Q�.��Let��E�f��b�A�e�the�eastern�v��9ertex�of��T�����1��*��.��let����W�R���b�A�e���the���v��9ertex�b�et��9w�een����P���and����V���(shared�b��9y��T�����2��ρ�and��T�����3��*��)�and��S�'��the�v��9ertex����Wb�A�et��9w�een��L�V���and��Q��(shared�b��9y��T�����4����and��T�����5��*��).�hXLet��M��0�b�e�the�gure�formed����Wb��9y��/these�v�e��.triangles.�Note�that�w��9e�ha�v�e�not�pro�v�ed��.that�triangles��T�����2�����W�and����T�����3��Ċ�ha��9v�e�a�common�\w�est"���v�ertex,��i.e.��their�shared�sides�ma�y���b�A�e�of����Wdieren��9t�Tlength,�and�the�same�go�A�es�for��T�����4��?��and��T�����5��*��.����d��src:1312 SevenTriangles.texBy�FLemma�1,�R8�
�f��=����R�=�2�and��T�����2��p��and��T�����3��p��ha��9v�e�Frigh�t�angles�at��R�>�,�R8and��T�����4�����W�and�z�T�����5�����ha��9v�e�righ�t�zangles�at��S���.�J�Then��P���and��Q��are�v��9ertices�of��M����,��Fas�are����W�E�u��and��the��w��9estern�v�ertices��of��T�����3��S�and��T�����5��*��.��Th�us��w�e�cannot��aord�to�insert����Wt��9w�o��triangles��b�A�et�w�een��T�����3��/��and���T�����4���(that�is,�/an��9ywhere��inside�the�angle�at��V����W�b�A�et��9w�een��)�T�����3�����and��*�T�����4��*��),���as�that�will�mak��9e�sev�en�altogether�and��*the�result�will����Wnot�)$b�A�e�)%a�triangle,�.as�it�m��9ust�con�tain�)%a�w�esternmost�)%v�ertex�in�)%addition�to����W�P�H��,��m�Q�,��nand��s�E��2�.�%Hence��tthere�is�just�one�triangle��T�����6����b�A�et��9w�een��T�����3����and��T�����4��*��.�%The����Wlongest��&side��%�c��of��T�����3�����(the�h��9yp�A�oten�use)��&is�shared�with��T�����6��*��,��band�since�no�other����Wtriangle��,can�b�A�e�inserted�b�et��9w�een��,�T�����3����and��T�����4��*��,��hthe�side�of��T�����6���shared�with��T�����3�����W�m��9ust��Falso��Eha�v�e�length��E�c�.��Similarly��:�,��|�T�����6�����m�ust�share��Eside��c��with��T�����4��*��;�؟but�then�����>�color push gray 0�����16��Y�	color pop����S��y�������>�color push gray 0�Y�	color pop���?������W�T�����6��ϝ�has���t��9w�o�t�w�o���sides�equal�to�the�h�yp�A�oten�use,���whic�h�is�a�con�tradiction.����WThis�Tcon��9tradiction�completes�the�pro�A�of�of�the�lemma.������R�color push gray 0��Lemma���5�	color pop������src:1328 SevenTriangles.tex�No��)�7�-tiling�c��ontains�a�non-strict��(vertex�of�typ�e�other�than��3���:�1����W�or�N<�2���:�1��or��2�:�2�.����W�src:1331 SevenTriangles.texPr��o�of�.���Let�;��V�I��b�A�e�;�a�non-strict�v��9ertex�in�a�7-tiling.���Then�for�some�in��9tegers����W�m�@
�and�@	�n�,�J�there�is�a�maximal�segmen��9t��S���con�taining�@
�V�NA�of�t��9yp�A�e��m����:���n�.���W��:�e����Wha��9v�e�&�m�c�+��n������7�since�there�are�only�&7�triangles�in�the�tiling.�N�Visualize��S����W�as��orien��9ted�in�the�north-south�direction,��@with��n��triangles�w�est�of��S�%�and����W�m�� �triangles��!east�of��S���.�
Let��M���b�A�e�the�conguration�of�triangles�in�the�nal����Wtiling��Sthat��Ttouc��9h��S���.�&nNo�more�than�t�w�o��Ttriangles�on�the�same�side�of��S����W�can�<ashare�a�<`common�v��9ertex�that�is�not�on��S���.���Hence�if��n��(or��m�)�is�three,����Wthen�KVat�least�four�triangles�m��9ust�o�A�ccur�on�KUthe�w�est�(or�east)�of��S���;�fWand�if����W�n��C�(or��m�)�is�four,���then�at�least�v��9e�triangles�m�ust��Bo�A�ccur�on�the�w�est�(or����Weast);�j3and���neither��n��nor����m��can�b�A�e�as�m��9uc�h�as���v�e,�1bsince�then���at�least����Wsev��9en�Ttriangles�w�ould�b�A�e�required�on�one�side�of��S���.����d��src:1339 SevenTriangles.texW��:�e��Bma��9y��Ac�hange�\east"��Aand�\w�est"�if��Anecessary�to�ensure��m�R���n�.����WSupp�A�ose�ގ�n��'�=��(4.�xSince�ޏw��9e�ha�v�e�pro�v�ed�ab�A�o�v�e�ޏthat��S�a��cannot�b�A�e�of�t��9yp�A�e����W4�`�:�`�1,���there���are�at���least�t��9w�o���triangles�east�of��S���,���and�as�remark��9ed�ab�A�o�v�e,����Wat��Yleast�v��9e��Xw�est��Yof��S���.�F��:�or�this�to�b�A�e�the�case,���the�triangles��T�����1����and��T�����2���on����Wthe��north��9w�est���m�ust�share�a���side�and�b�A�oth�ha��9v�e�righ�t�angles���where�they����Wmeet�r��S���,���and�the�r�same�for�triangles��T�����3���?�and��T�����4���on�the�r�south��9w�est,���and�for����Wtriangles����T�����5���?�and��T�����6���@�on�the���east,��*and�then��T�����7���m��9ust�share�v�ertex��V����on��S����W�with�з�T�����2���]�and��T�����3��*��.��That�жmeans�that�the�largest�angle��
�TD�is�a�righ��9t�angle,��pand����Wthe��Ksev��9en-triangle�conguration�has�v�ertices�at��Lthe�endp�A�oin�ts�of��S��P�and�at����Wleast��one�v��9ertex�east��of��S����and�at�least�one�v��9ertex�w�est�of��S���,��and�hence�is����Wnot�Ta�triangle.�pHence��n����=�4�Tis�not�p�A�ossible.����d��src:1348 SevenTriangles.texSupp�A�ose�y��n�8��=�8�3.�H�W��:�e�ha��9v�e�y�pro�v�ed�y�ab�o�v�e�y�that�y�t�yp�e�y�3�8�:�2�y�is�imp�A�ossible.����WW��:�e��sno��9w��rconsider�t�yp�A�e��r3�G�:�3.�c�But�as��sremark��9ed�ab�A�o�v�e,���this��rw�ould�require����Wfour���triangles�east�of��S���and�four�triangles�w��9est�of��S���,��making�more�than����Wsev��9en,�Tso�3���:�3�Tis�imp�A�ossible.�pThat�completes�the�pro�of�of�the�lemma.����d��src:1352 SevenTriangles.texThe�T�follo��9wing�T�gures�sho�w�T�some�p�A�ossible�congurations�in�whic��9h�a����Wmaximal�C9segmen��9t�C8of�t�yp�A�e�2��C:�1�C9could�o�ccur.��In�C8these�gures,�N���E�has�to�b�A�e����Was���sho��9wn,��
either���R�=�6�or��arctan������P�� ��1�� ���&�fe�����2�����$�z�,��	so�that���2�a�b��=��c����or�2�a�b��=��b�.���In���the�rst����Wt��9w�o�q�gures,����
��_�has�q�to�b�A�e�a�righ��9t�angle.�1�In�the�third�gure,����
��_�and�����ha��9v�e����Wone�Tdegree�of�freedom;�the�gure�illustrates�the�case��
��=���80�degrees.������>�����color push gray 0�v
���乍��M���color push gray 0�Figure�UU7:�q�Tw���o�three-triangle�congurations�	color pop���eG��src:1360 SevenTriangles.tex����h5:���JP���g�Y��#Q���g�Y��R%V������R%W��������E��s��q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.94214 42.67912 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.94214 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.94214 85.35825 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.94214 0.0 moveto 73.94214 85.35825 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.94214 0.0 moveto 110.88391 64.01868 L 0 setlinecap stroke  end �w" tx@Dict begin STP newpath 0.8 SLW 0  setgray  110.88391 64.01868 moveto 73.94214 85.35825 L 0 setlinecap stroke  end ���������IP�����NQ������ٖ�V����y��R%W���)�˟��IE�����q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 85.35826 42.67912 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 85.35826 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 85.35826 85.35826 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  85.35826 0.0 moveto 85.35826 85.35826 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  85.35826 0.0 moveto 128.03738 85.35826 L 0 setlinecap stroke  end �w" tx@Dict begin STP newpath 0.8 SLW 0  setgray  85.35826 85.35826 moveto 128.03738 85.35826 L 0 setlinecap stroke  end �����	color pop�����>�color push gray 0����17��Y�	color pop����j��y�������>�color push gray 0�Y�	color pop���?���P
��>�����color push gray 0�v
���乍��Q?��color push gray 0�Figure�UU8:�q�Tw���o�v�e-triangle�congurations�	color pop���eG��src:1369 SevenTriangles.tex����hk韪�JP���g���ٖ�V���g���D�Q����E����E����ğD�R����ğ���S�����ٖ�W��s��q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 85.35825 moveto 73.94214 85.35825 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.94214 42.67912 L 0 setlinecap stroke  end �g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 73.94214 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.94214 0.0 moveto 73.94214 85.35825 L 0 setlinecap stroke  end �g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 0.0 85.35825 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 73.94214 42.67912 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 85.35825 moveto 73.94214 42.67912 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.94214 0.0 moveto 110.88391 64.01868 L 0 setlinecap stroke  end �w" tx@Dict begin STP newpath 0.8 SLW 0  setgray  110.88391 64.01868 moveto 73.94214 85.35825 L 0 setlinecap stroke  end �������UʟD�R�����D�ٖ�W����Uʟ���S�����'���P���'誟��!E�����U�D�Q�����U�ٖ�V�����r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 85.35826 moveto 128.03738 85.35826 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 85.35826 42.67912 L 0 setlinecap stroke  end �g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 85.35826 0.0 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 85.35826 42.67912 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 85.35826 moveto 85.35826 42.67912 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  85.35826 0.0 moveto 85.35826 85.35826 L 0 setlinecap stroke  end �g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 0.0 85.35826 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  85.35826 0.0 moveto 128.03738 85.35826 L 0 setlinecap stroke  end �����	color pop���䍑>����color push gray 0�v䍟��㍍�O�f�color push gray 0Figure�UU9:�q�Tw���o�four-triangle�congurations�	color pop���e��src:1378 SevenTriangles.tex����l���NQ���m�����)P��������g>E���m����R%V���4�� LS���5Ο��'R��s��q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  80.21092 0.0 moveto 80.21092 85.35826 L 0 setlinecap stroke  end �w" tx@Dict begin STP newpath 0.8 SLW 0  setgray  120.31638 14.59671 moveto 80.21092 85.35826 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  120.31638 14.59671 moveto 80.21092 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 71.87384 moveto 80.21092 85.35826 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 71.87384 moveto 80.21092 42.67912 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 29.19472 moveto 80.21092 42.67912 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 29.19472 moveto 80.21092 0.0 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 71.87384 moveto 0.0 29.19472 L 0 setlinecap stroke  end ������	'_��NQ���
o���)P���3*��=E���
o��R%V�������� LS�����G���'R����`�q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  80.21092 0.0 moveto 80.21092 85.35826 L 0 setlinecap stroke  end �w" tx@Dict begin STP newpath 0.8 SLW 0  setgray  120.31638 70.76154 moveto 80.21092 85.35826 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  120.31638 70.76154 moveto 80.21092 0.0 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 71.87384 moveto 80.21092 85.35826 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 71.87384 moveto 80.21092 42.67912 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 29.19472 moveto 80.21092 42.67912 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 29.19472 moveto 80.21092 0.0 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 71.87384 moveto 0.0 29.19472 L 0 setlinecap stroke  end �����	color pop������R�color push gray 0��Lemma���6�	color pop������src:1386 SevenTriangles.tex�Supp��ose�Nthat�Na��7�-tiling�c��ontains�a�maximal�se��gment�of�typ��e����W�2��~:��1�.���Then�wothe�tiling�wnc��ontains�one�of�the�six�c��ongur�ations�woshown�in�the����Wpr��e�c�e�ding�!Dgur�es.��T��J�o�state�!Ethe�c�onclusion�without�r�efer�enc�e�to�a�gur�e:����Wthe���smal�x�lest�angle������of�the�tile�is���R�=�6��or���arcsin������P��4��1��4���&�fe�����2�����#~�,��hso��2�a�a��=�a��c��or��2�a��=�a��b�;����Wthe�3maximal�3se��gment�has�length��2�a�,�lRwith�the�non-strict�vertex��V�AS�at�its����Wmidp��oint,�N<and�one�of�the�fol�x�lowing�holds.����d��src:1393 SevenTriangles.tex(i)�̷3�triangles�̶me��et�at�the�non-strict�vertex,��two�of�them�having�a�right����Wangle�N<ther��e.�@Se�e�Fig.�@8.����d��src:1396 SevenTriangles.tex(ii)���5���triangles�me��et�at�the�non-strict�vertex.�*�Denoting�the�maximal����Wse��gment���by����P�H�Q�,���with�midp�oint��V�8�,���the���triangles�\west"�of��P�H�Q��with�vertic��es����Wat���P���and��Q��have�right��angles�at��P��and��Q�,��and�angle���D&�at���V�8�,�and�the�other����Wtwo�N<triangles�west�of��P�H�Q��have�angle���P�at��V�8�.�@Se��e�Fig.�9.����d��src:1401 SevenTriangles.tex(iii)��e3�triangles�on��done�side�of�the�maximal�se��gment�shar�e��ethe�vertex��V�8�,����Wwith�@�the�midd�x�le�one�@�(the�one�that�do��es�not�shar�e�a�@�side�with�the�maximal����Wse��gment)��having��angle���`�at��V�8�,�-0and�sharing�another�vertex�with�e��ach�of����Wthe�Rother�two�triangles�with�which�it�shar��es�vertex�Q�V�8�.�-`Mor�e�over,�!�e�ach�two����Wadjac��ent�atriangles�`of�the�thr��e�e�aon�one�side�of�the�maximal�se��gment�form�a����Wp��ar�al�x�lelo�gr�am.�@Se�e�N<Fig.�10.����d��src:1406 SevenTriangles.texNote��Ithat��Jin�c��ases�(i)�and�(ii),��the�tile�is�a�right�triangle,��while�in�c��ase����W(iii)�N<that�is�not�asserte��d.����W�src:1409 SevenTriangles.texPr��o�of�.�?The���previous���lemmas�ha��9v�e�ruled���out�all�p�A�ossible�t��9yp�es���of�maximal����Wsegmen��9ts��(except��)3�
}:�1,�22�:�2,�and��)2�:�1.���W��:�e��(consider�the�p�A�ossible�con-����Wgurations�r�in�r�whic��9h�a�maximal�segmen��9t�has�t��9yp�A�e�2�-v:�-u1.�4�F��:�or�con�v�enience����Wof�xdescription,�
plet�us�worien��9t�triangle��AB�r�C��`�so�the�maximal�segmen��9t��P�H�Q��is����Wnorth-south,��with��Mt��9w�o��Ntriangles�on�the�w��9est�and�one�on�the�east.�	nBecause�����>�color push gray 0�����18��Y�	color pop�����	�y�������>�color push gray 0�Y�	color pop���?������W�there���is���just�one�triangle�on�the�east,���the�length�of��P�H�Q��is�either��b��or��c�.����WThe���t��9w�o���triangles�on�the�left�m��9ust�divide�the�maximal�segmen��9t�equally��:�,����Wb�A�ecause���if�they�did�not,���then�the�t��9w�o���segmen�ts�w�ould�ha�v�e�lengths��a��and��b����W�and���their���sum�(the�side�of�the�one�triangle�on�the�righ��9t)�w�ould���necessarily����Wb�A�e��l�c�,�Țbut�of�course��k�a�Nh�+�Ng�b���<�c�.��xIt��lfollo��9ws�that�the�t��9w�o��lsegmen�ts�ha�v�e��llength����W�a�,�`�and�Q�2�a��f�=��e�b�Q��or�2�a��=��e�c�,�`�since�Q�if�the�segmen��9ts�had�length��b��instead�of��a�,����Ww��9e�Kw�ould�ha�v�e��c��l�=�2�b����a�2�+��b�,�X�so�a�triangle�with�sides��a�,�X��b�,�and��c��w��9ould����Wb�A�e�Timp�ossible.����d��src:1422 SevenTriangles.texW��:�e�� consider�the��\minimal�conguration"��M���con��9taining�all�the��T�����i��В�with����Wa�e�v��9ertex�at��V�8�.�
VHo�w�man�y�triangles�will��M�^��con�tain?�
VIt�con�tains�at�least����Wthree.�pW��:�e�Twill�analyze�the�p�A�ossibilities.����d��src:1425 SevenTriangles.texLet��ithe��jdirection�of��P�H�Q��b�A�e�\north-south"�with��P�L�at�the�north.��Let��T�����1�����W�and�j�T�����2�����b�A�e�the�j
w��9est�triangles�and��T�����3�����the�east�triangle�with�sides�on��P�H�Q�,����Wand�Ҷ�V����the�midp�A�oin��9t�of�ҷ�P�H�Q�,�the�shared�Ҷv�ertex�of�ҷ�T�����1���]�and��T�����2��*��.�T�If��T�����1���]�and����W�T�����2���w�share���a�side,�/then�the�largest�angle��
�V^�is�a�righ��9t�angle,�so��T�����1���w�and��T�����2�����W�m��9ust�y�also�y�share�their�w��9est�v�ertex��W��(at�distance�y��b��from��P�H�Q�).�IzThen�w�e����Wha��9v�e�%�2�a����=��b��or�2�a��=����c�,�)�leading�to�the�%�p�A�ossibilities�listed�in�part�(i)�of�the����Wconclusion�Tand�illustrated�in�Fig.�p8.����d��src:1432 SevenTriangles.texW��:�e�)therefore�ma��9y�assume�that��T�����1��I��and��T�����2��I��do�not�share�a�side,�a�and����Wat��@least��Aone�additional�triangle��T�����4�����shares�their�common�v��9ertex��V��x�at�the����Wmidp�A�oin��9t��<of��P�H�Q�.�Let��S�dA�b�e�the�w��9est��;v�ertex��<of��T�����1����and��R��z�the�w��9est�v�ertex�of����W�T�����2��*��,���and��H�E�cz�the��Ieast�v��9ertex�of��T�����3���.�
�Then�angles��P�H�S��V���and��V�8R�>Q��are�b�A�oth�����,����Wsince�Tthey�are�opp�A�osite�side��a�.����d��src:1437 SevenTriangles.texAssume�^Arst�that�^@there�is�exactly�one�triangle��T�����4�����b�A�et��9w�een�^A�T�����1�����and��T�����2�����W�sharing�H�v��9ertex��V�8�.��One�of�the�sides�of��T�����4��s��lying�along�H��S��V�W�or��R�>V��m��9ust�b�A�e����W�b�0��or�0��a��and�since�the��a��sides�of��T�����1��[l�and��T�����2���lie�on�0��P�H�Q�,�7�the�sides��S��V�>��or��R�>V����W�m��9ust��eac�h��b�A�e��b��or��c�.��Assume,�[for�pro�of�b��9y�con�tradiction,�\that��triangle��T�����4�����W�do�A�es�U�not�share�w��9est�v�ertices�with��T�����1���B�and��T�����2��*��,�e�i.e.��Eit�is�not�triangle��S��V�8R�>�.����WThen���one�of���its�v��9ertices�lies�on��S��V����or�on��R�>V�8�,���since�the��b��side�of��T�����4���n�cannot����Wb�A�e�{�longer�than�{�the��b��or�the��c��sides�of��T�����1���7�and��T�����3��*��.�O&In��9terc�hanging�{�\north"����Wand���\south"�if�necessary��:�,��4w��9e�can���assume�that�the�north�v�ertex��X�c��of��T�����4�����W�lies��on���S��V�8�,�"�and�hence�is�a�non-strict�v��9ertex.��zSince��S��V����m�ust��b�A�e�larger����Wthan�?S�X��[V�8�,�I�w��9e�ha�v�e�?Reither��S��V���=����c��and��X��[V���=����b�,�I�or��S��V��=����c��and�?R�X��[V��=����a�,����Wor�U��S��V�4�=����b��and��X��[V��=����a�.��jThe�maximal�U�segmen��9t�of�this�non-strict�v��9ertex����Whas��one�end�at��V��M�and�extends�w��9est�w�ard��along���S��V�8�.�By�Lemma�5,���its�t��9yp�A�e����Wm��9ust�Tb�A�e�2���:�1�Tor�3���:�1�Tor�2���:�2.����d��src:1451 SevenTriangles.texCase�2Z1,�_�angles��S��V�8P�{=�and��R�>QV�@��are�2Yequal.���Since�these�2Zare�corresp�A�onding����Wangles���made���b��9y�the�transv��9ersal��P�H�Q��to��S��V��*�and��R�>Q�,��lines��S�V��*�and��R�>Q����are����Wparallel.�S�Then�'�the�'�alternate�in��9terior�angles��S��V�8R�7��and��V�R�>Q�'��are�equal.����WAngle���V�8R�>Q����=�����,��Bsince�it�is�opp�A�osite��V�8Q��=��a�.��Therefore�angle��S��V�8R���=�����.����WTherefore����X��[V����=����b��or��X�V����=����c�;���but�since��X�f!�lies�on��S��V����and��X�J��6�=��S�4��w��9e����Wha��9v�e����S��V�^��=�P��c����and��X��[V��=�P��b�.�s�If�the���t��9yp�A�e�of�the�maximal�segmen��9t�of��X�<!�is����W2�=�:�=�1,��mw��9e�|�ha�v�e�|��a�S�+��b��=�=��c�,��nwhic��9h�|�is�imp�A�ossible�|�since��a�S�+��b�>�=�c�,��nor�|�2�b��=��c�,����Wwhic��9h���is���imp�A�ossible�since�2�b��t>�a�w��+��b��t>�c�.��If���it�is�3��t:�1���w��9e�ha�v�e���3�b��t�=��c�,����Wor�[�a�<��+�2�b��=��c�,�l�or�2�a��+�<��b��=��c�,�l�all�of�whic��9h�[are�imp�A�ossible�since��a�<��+�<��b�>�c�.����WBy�8Lemma�85,�@�the�only�remaining�p�A�ossibilit��9y�for�the�t��9yp�A�e�of�the�maximal����Wsegmen��9t��'of��X����is�2��&:�2.���If��(�
�n��is��'not�a�righ�t�triangle,� �then�there�m�ust�b�A�e����Wthree��triangles��on�eac��9h�side�of�the�maximal�segmen��9t,���making�six�triangles,����Wwhic��9h��together�with���T�����2��H�and��T�����3���is�more�than��sev��9en.��WHence��
�k.�is�a�righ�t����Wangle.���Since��c�S��V���=����c�,��'angle��d�S�P�H�V�›�is�a��drigh��9t�angle�and�angle��S��V�8P����=�����x,�.�����>�color push gray 0�����19��Y�	color pop�����@�y�������>�color push gray 0�Y�	color pop���?������W�Hence�vPangle��R�>QV�A��=�3j���|�and�angle�vQ�R�V�8Q��is�a�righ��9t�vQangle.�?dSince��X��[V�A��=�3j�b�,����Wthe��righ��9t�angle�of��T�����4����is��at��X��[�.���Therefore�the�south�side�of��T�����4����is��c�,��and����Wextends��Xw��9est��Wof��R����on��R�>V�8�.�r{W��:�e�no��9w�ha�v�e��Wa�third�non-strict�v��9ertex�at��R�>�.����WThe��exterior��angle�at��R��C�is�more�than���R�=�2,���so�w��9e�will�need�at�least�t��9w�o����Wmore��triangles��to�sharing�v��9ertex��R���to�b�A�e�placed�south�of��R�>V�8�.��But�w��9e����Wm��9ust���also���place�at�least�t��9w�o�more���triangles�with�sides�on�line��S��V�8�,��=since����Wthe��}maximal�segmen��9t��|of��X����has�t�yp�A�e�2�_:�2.���That��|mak�es��}eigh�t�triangles����Waltogether,�Twhic��9h�is�more�than�sev�en.�pThat�disp�A�oses�of�Case�1.����d��src:1473 SevenTriangles.texCase��2,�/�angles��S��V�8P�?��and��R�>QV�I�are�not�equal.���Since�triangles��S�P�H�V����W�and�\"�R�>QV�jZ�are�congruen��9t�and��P�H�V���=���V�8Q��=��a�,�m�w�e�\"m�ust�ha�v�e�angles��S��V�8P����W�and��R�>V�8Q��equal;�
wand�they�m��9ust�b�A�oth�b�e�equal�to���x,�,�Jsince�the�angle�of��T�����4�����W�at����V����is�at���least�����,���and�2�
��E�+�i���xL>�v��k�+�i������+��
���=���R��.��.Then���the���angle�of��T�����4�����W�at�?K�V�M��cannot�b�A�e�?L�
����,�I�since�2���_�+�*2�
�[F>�׸�R��.��VIf�it�is���A�then�w��9e�ha�v�e�?L2���^�+�*3���~�=�׸��R��,����Wwhic��9h�|�w�ould�mak�e����^�=�>2�
�V�so�|�angles��S��V�8P�ū�and��R�>QV���w�ould�b�A�e�equal�and����WCase��n1��ow��9ould�apply��:�.���Therefore�it�is�����and�w��9e�ha�v�e��o3��	��=�����R��.���Then�side��n�X��[V����W�of��triangle��T�����4��;J�m��9ust�b�A�e��a�,��since�it�cannot��b�e��c�,��b�ecause�it�is�less�than��S��V�8�.����WConsider��8the��7t��9yp�A�e�of�the�maximal�segmen��9t�of��X��[�.�Assume,��0for�pro�A�of�b��9y����Wcon��9tradiction,�"that��it��is�2���:�1.��Then��2�a��=��c��and���X����is�the�midp�A�oin��9t�of��S��V�8�.����WSince�u���	��=�����R�=�3,���the�equation�2�a��=��c��implies��
��Z�is�a�u�righ��9t�angle�and������=���R�=�6.����WThen�BVthe�BWside�of��T�����4��l��on�line��R�>V�P��m��9ust�b�A�e�the��c��side,�M�since�if�the�v��9ertex�of����W�T�����4���=�on����R�>V����lies�b�A�et��9w�een����R����and����V�8�,���the�distance�from��X�Q��to�that�v��9ertex�is����Wless���than����c�.���Hence��T�����4���z�has��R���for�its�south��9w�est���v�ertex.���No�w�w�e�ha�v�e���a�four-����Wtriangle��wconguration�with��vparallel�north�and�south�b�A�oundaries��S��P��Z�and����W�R�>Q�,�B�a�9�conca��9v�e�9�exterior�v��9ertex�at��X��[�,�B�and�a�righ��9t�angle�at�the�east�v��9ertex����W�E��2�.��If��three��additional�triangles�are�added�sharing�v��9ertex��X��[�,��the�resulting����Wconguration���of�sev��9en�triangles�will�not���b�A�e�a�triangle,��Xsince�it�has�three����Wv��9ertices����P�H��,��3�Q�,��4and����E�&��and�more�to�the�w��9est�of��P�H�Q�.��VIf�t��9w�o���triangles�are����Wadded���sharing�v��9ertex��X��[�,���they���m�ust�not�b�A�e�placed�so�as���to�create�new�non-����Wstrict�=-v��9ertices,�hhas�that�=,w�ould�require�placing�a�=,sev�en�th�triangle�w�est�=,of��P�H�Q�,����Wlea��9ving���at�least�four�v�ertices.���Therefore�if�t�w�o�more�triangles�are�added����Wsharing���v��9ertex��X��[�,� Qthey�share�a�w�est�v�ertex����W�3��and�are�triangles��S��X��[W����W�and���R�>X��[W�H��.�The�resulting��six-triangle�conguration�is�con��9v�ex�and��has�six����Wv��9ertices.��Placing���one�more�triangle���can�decrease�the�n�um�b�A�er���of�v�ertices����Wof��3a��4con��9v�ex�conguration�b�y�at��4most�one,�mso�this�conguration�cannot�b�A�e����Wcompleted�-to�-a�7-tiling.�c�Hence�the�conca��9v�e�exterior�-v�ertex�at��X��k�m�ust�b�A�e����Wlled���b��9y���just�one�triangle.��If�this�triangle��T�����5���x�is�not��S��X��[R���then�its�w��9est����Wv��9ertex�Aa�W��C�lies�on�A`�S��V�O��extended,�k�and�it�has�side��b��along��S��V�8�.���Its�south�v��9ertex����W�U��lies�2on�1�R�>X��[�.�dNew�conca��9v�e�exterior�angles�2are�created�at��S��6�and��U����,�keac��9h����Wof�p�whic��9h�p�is�more�than���R�=�2,���and�hence�eac��9h�will�require�placing�at�least�t��9w�o����Wmore�vBtriangles�with�v��9ertices�vC�S��G�and��U�o&�resp�A�ectiv�ely��:�.�?;But�that�will�require����Wa�1Htotal�1Gof�9�triangles.�pKHence�triangle��T�����5��[��m��9ust�b�A�e�triangle��S��X��[R�>�.�pKW��:�e�no��9w����Wha��9v�e��a��v�e-triangle�conguration�including��rectangle��S��P�H�QR�9�and�triangle����W�P�H�QE��2�.��1Either��>�QE��+�=�\��b��or��?�P�E��*�=�\��b�.��1Supp�A�ose,���for�pro�of�b��9y�con�tradiction,����Wthat���QE���=����b�.�Then�it��cannot�b�A�e�that�b�A�oth��R�>S�X�and��QE�X>�lie�on�sides�of�the����Wnal�/�triangle,�6Gsince�the�/�area�to�b�A�e�lled�south�of��R�>Q��w��9ould�require�more����Wthan��t��9w�o��
triangles.�ǞWith�only�t��9w�o�more�triangles��
a�v��|railable,�1�w�e�cannot����Wcreate���more���non-strict�v��9ertices.�FConsider�placing�a�triangle��T�����6���B�south�of����W�QE��2�.�Then�i�it�i�m��9ust�share�v��9ertices��Q��and��E��2�.�If�w��9e�do�not�place�the�righ��9t����Wangle��at���E����then�there�will�b�A�e�a�conca��9v�e��exterior�v�ertex��at��Q��that�will����Wrequire�h�t��9w�o�h�more�triangles�to�ll,�}�con��9tradiction.��Hence��T�����6���f�m�ust�h�ha�v�e�its�����>�color push gray 0�����20��Y�	color pop�������y�������>�color push gray 0�Y�	color pop���?������W�righ��9t�3�angle�at��E��2�.��4No�w�w�e�ha�v�e�a�six-triangle�con�v�ex�conguration�with�v�e����Wv��9ertices.��This��cannot�b�A�e��completed�to�a�7-tiling�since�adding�one�triangle����Wto��a�con��9v�ex��conguration��can�reduce�the�n��9um�b�A�er��of�v��9ertices�b�y��at�most����Wone.�Hence��A�QE�sr�is�one��@of�the�sides�of�the�nal�triangle.�Placing�a�triangle����W�T�����6���`�north���of��P�H�E�=��without�creating���a�new�non-strict�v��9ertex�w�ould�require����Wthat����T�����6���9�ha��9v�e�side��a��along����P�H�E��2�;��)that�w��9ould�create�a�conca�v�e���exterior�v�ertex����Wat�F;�P���greater�F<than���R�=�2,�Ruwhic��9h�could�not�b�A�e�lled�with�one�more�triangle.����WHence�X��P�H�E���is�X�also�one�of�the�sides�of�the�nal�triangle.��Then�there�m��9ust����Wb�A�e�Ja�Ktriangle��T�����6��/��north�of��S��P�N-�whose��c��side�lies�on��P�H�E��|�extended.�But�no��9w����Wt��9w�o��triangles��ha�v�e�sides��on�line��R�>S���,��1and�w��9e�ha�v�e��already�seen�that�not����Wb�A�oth�,@�R�>S��D�and��QE��q�can�b�e�,?sides�of�the�nal�triangle.�a3So�at�least�t��9w�o�,@more����Wtriangles�``will�`ab�A�e�required�w��9est�of��R�>S���,���but�w��9e�ha�v�e�only�one�`amore�a�v��|railable.����WThis�Tcon��9tradiction�sho�ws�that��QE���6�=����b�.����d��src:1521 SevenTriangles.texTherefore����P�H�E��2�=�|�b�.��Then�it�cannot���b�A�e�that�b�oth��R�>S�$��and����P�H�E�%�lie�on����Wsides��Sof�the��Tnal�triangle,��Ssince�the�area�to�b�A�e�lled�north�of��S��P��6�w��9ould����Wrequire�!more�than�!t��9w�o�!triangles.�?�With�only�t��9w�o�!more�triangles�a��9v��|railable,����Ww��9e�1cannot�2create�more�non-strict�v�ertices.�eConsider�placing�a�2triangle��T�����6�����W�north��Qof��P�P�H�E��2�.��oThen�it�m��9ust�share�v��9ertices��P��4�and��E��2�.��oIf�w��9e�do�not�place�the����Wrigh��9t��uangle��vat��E���then�there�will�b�A�e�a�conca��9v�e�exterior��uv�ertex�at��P��Y�that�will����Wrequire�h�t��9w�o�h�more�triangles�to�ll,�}�con��9tradiction.��Hence��T�����6���f�m�ust�h�ha�v�e�its����Wrigh��9t�3�angle�at��E��2�.��4No�w�w�e�ha�v�e�a�six-triangle�con�v�ex�conguration�with�v�e����Wv��9ertices.��This��cannot�b�A�e��completed�to�a�7-tiling�since�adding�one�triangle����Wto��a�con��9v�ex��conguration��can�reduce�the�n��9um�b�A�er��of�v��9ertices�b�y��at�most����Wone.��Hence��_�P�H�E�u��is�one�of��`the�sides�of�the�nal�triangle.��Placing�a�triangle����W�T�����6���9�south���of��QE�:��without�creating���a�new�non-strict�v��9ertex�w�ould�require����Wthat��t�T�����6����ha��9v�e�side��s�a��along��QE��2�;�ƿthat�w�ould�create��sa�conca�v�e�exterior�v�ertex����Wat�D4�Q�D5�greater�than���R�=�2,�O�whic��9h�could�not�b�A�e�lled�with�one�more�triangle.����WHence�V��QE���is�also�one�of�the�sides�V�of�the�nal�triangle.��Then�there�m��9ust����Wb�A�e���a���triangle��T�����6����south�of��R�>Q��whose��c��side�lies�on��QE�t �extended.�MBut�no��9w����Wt��9w�o��triangles��ha�v�e�sides��on�line��R�>S���,��1and�w��9e�ha�v�e��already�seen�that�not����Wb�A�oth�-��R�>S����and��P�H�E���can�b�e�sides�of�the�nal�triangle.�e�So�at�least�t��9w�o�-�more����Wtriangles�``will�`ab�A�e�required�w��9est�of��R�>S���,���but�w��9e�ha�v�e�only�one�`amore�a�v��|railable.����WThis�Z*con��9tradiction�Z+disp�A�oses�of�Case�2.���That�in�turn�completes�the�pro�A�of����Wb��9y�Tcon�tradiction�that�triangle��T�����4��?��is�triangle��S��V�8R�>�.����d��src:1542 SevenTriangles.texNo��9w�Tthat�Tw�e�kno�w��T�����4��~��is�Ttriangle��S��V�8R�>�,�z�w�e�again�ha�v�e�cases�Tto�consider.����WCase�s�1,��pangles��S��V�8P����and��R�>QV��	�are�equal.�7�Since�these�s�are�corresp�A�onding����Wangles�zmade�yb��9y�the�transv�ersal�y�P�H�Q��to��S��V� ��and��R�>Q�,�Q�lines��S�V� ��and��R�>Q����W�are��gparallel.�!Then�the�alternate�in��9terior��fangles��S��V�8R���and��V�R�>Q��are�equal.����WAngle���V�8R�>Q����=�����,��Bsince�it�is�opp�A�osite��V�8Q��=��a�.��Therefore�angle��S��V�8R���=�����.����WSince�v�angles��R�>S��V��(�and��V�8QR��.�are�v�opp�A�osite�side��R�V��(�in�their�resp�A�ectiv��9e����Wtriangles,�Ҳthey���are�equal.���Since����V�8Q��and��S��R���are�opp�A�osite�angle�����,�Ҳthey����Ware��}equal.���Hence�sides��~�S��V����and��R�>Q��}�are�also��~equal.���That�mak��9es��}�S�V�8QR����W�a���parallelogram.�mThis���is�the�conguration�describ�A�ed�in�part�(iii)�of�the����Wlemma,�Tso�w��9e�are�nished�with�Case�1.����d��src:1552 SevenTriangles.texCase��C2,��angles��S��P�H�V��{�and��R�>QV��are��Bequal.�Then�since�angle��P�H�S��V����=�������,����Wangles����S��P�H�V���and��R�>QV���are���equal�either�to������or�to��
����.�g�Then�angles��S��V�8P����W�and�r-�R�>V�8Q�r,�are�also�equal�(either�to��
����or�to���x,�).�2�They�cannot�b�A�e�equal�to����W�
����,�ҟb�A�ecause���in�that���case�it�w��9ould�not�b�e�p�ossible�to�place�ev��9en�one�triangle����W�T�����4�����b�A�et��9w�een�t��T�����1���and�t��T�����2��*��,���since���O��+�M�2�
�����1(���+����%�+�M��
����=�1(��R��.�;THence�t��S��V�8P����and�����>�color push gray 0�����21��Y�	color pop����ˠ�y�������>�color push gray 0�Y�	color pop���?������W�R�>V�8Q�η�are�equal�to���x,�.�H�Then�2��F��plus�angle��S��V�R���equals���R��.�H�Since�angles����W�S��P�H�V����and��t�R�>QV��are�b�A�oth��sequal�to��
��and��S��P��W�and��R�>Q��are�b�A�oth�equal�to����W�b�,��
�S��R��&�is���parallel�to��P�H�Q�,�and�triangle��S��V�8R��&�is�isosceles,�with�b�A�oth�sides����W�S��V�}�and�n��R�>V��equal�n�to��c�.�)*Hence�angles��S��R�>V�}�and��R�S��V�}�are�n�b�A�oth�equal�to����W�
����.�VAngles�(��S��R�>V�6��and�(��R�V�8Q��are�alternate�in��9terior�(�angles�of�the�transv�ersal����W�R�>V�(�of��parallel��lines��S��R�*�and��P�H�Q�,��so�they�are�equal;�but�angle��R�>V�8Q��3�=���x,�,����Wso��R�S��R�>V�RP�=�D���~�also.�]jThen�angle��S�R�>V����is�opp�A�osite�side��c��and�hence�equals����W�
����;��hence������=����
��.�'9Hence�the�triangles���T�����i��E^�are�isosceles�with�2�a����=��b��=��c�,��and����Wtriangle��"�S��V�8R�_�is�congruen��9t��!to�the��T�����i��,r�,�(and�hence�is�a�fourth�triangle��T�����4�����W�b�A�elonging�Tto�the�tiling.�pIn�this�case�conclusion�(iii)�of�the�theorem�holds.����d��src:1568 SevenTriangles.texThat�completes�the�pro�A�of�of�the�lemma�in�case�there�is�only�one�triangle����Wb�A�et��9w�een�T�T�����1��?��and��T�����2��*��.����d��src:1571 SevenTriangles.texW��:�e�rmstill�ha��9v�e�rlto�rmconsider�the�case�in�whic��9h�there�are�t��9w�o�rmor�more����Wtriangles,����T�����4�����(with�]a�]side�on��S��V�8�)�and��T�����5�����(with�a�side�on��R�>V�8�),���and�p�A�ossibly����Wstill�Cmore�Ctriangles,��nb�A�et��9w�een��T�����1��m��and��T�����2��*��.��|Changing�south�Cand�north�if����Wnecessary��:�,�\�w��9e�NPcan�NOassume�angle��P�H�QE�s��=������.��cAngle��QP�E�с�is�NPeither����|�or��
����.����WIf����
�z��is�not�a���righ��9t�angle,��
or�if��
��is�a�righ��9t�angle�but����S��P�H�E�z-�is�not�a�straigh�t����Wangle,�nthen��5this��65-triangle�conguration�has�v��9ertices�(at�least)�at��S���,�n�P�H��,����W�E��2�,����Q�,���and����R�>�,�and�at���least�one�more�v��9ertex��W��y�of��T�����4��*��.�y3If��W��is�a�conca��9v�e����Wv��9ertex��pit�will��oha�v�e�to�b�A�e�remo�v�ed�b�y�placing��o�T�����6����with�a�v�ertex�at��o�W�H��.��That����Wwill�n�lea��9v�e�at�least�n�v�e�other�v�ertices,��all�con�v�ex;��cthat�cannot�b�A�e�reduced����Wto�oHthree�b��9y�placing�one�more�triangle;��Ahence��W��+�is�not�a�conca�v�e�v�ertex.����WUnless�f�W�KI�o�A�ccurs�on�the�line��S��R�>�,�/w��9e�then�ha�v�e�six�con�v�ex�v�ertices,�/whic�h����Wcannot�*eb�A�e�*dreduced�to�three�b��9y�placing�t��9w�o�more�*etriangles.�[�The�only�w��9a�y����W�W����can�=�o�A�ccur�=�on��S��R�N�is�if�there�are�t��9w�o�triangles�=�b�A�et�w�een��T�����1��h��and��T�����2��*��,�Hand����Wthey��share��v��9ertex��W���and�ha��9v�e��a�righ�t��angle�there.��In�that�case��
���is�a�righ��9t����Wangle.�q�Hence�1��
��D�m��9ust�1�b�A�e�a�righ�t�angle,�8�and�one�1�of�t�w�o�cases�holds:�U4either����W2�a����=��c���and��angle��QP�H�E����=�����x,�,�&Nso������=�����R�=�6�and���u��=���R�=�3,�&Nand��there�are����Wt��9w�o�/Ytriangles�sharing�a�v��9ertex�at�/Zthe�midp�A�oin�t�of��S��R�?��and�another�v�ertex����Wat����V�8�,���or��QP�H�E���=����
����,�and��S��P�H�E�4��is���a�straigh��9t�angle.���Then�2�a����=��b�,���rather����Wthan�d2�a����=��c�,�aand������=��arctan������P����1�����&�fe�����2�����#��.��No��9w�consider�the�angles�at��V�8�.�The�angle����Wof�`��T�����1���Z�and��V�n��is�`���x,�.���If��T�����2���has�angle��
��A�there,�s�that�`�lea��9v�es�`�ro�A�om�for�only�one����Wtriangle��b�A�et��9w�een�them,�_�with��angle�����;���hence��T�����2��HB�has�angle������at��V�8�.�5DW��:�e����Wha��9v�e����u��=�����R�=�2���������arctan��V�(1�=�2),�&0whic�h�is�ab�A�out��63.43�degrees.��MHence�an�y����Wtriangles�\�b�A�et��9w�een�\��T�����1�����and��T�����2�����ha�v�e�angle���^��at��V�8�,�n�since����#�is�to�A�o�big�\�to�t.����WThe�1Qangle�to�b�A�e�1Rlled�is���s��� ��2��8��=��l2����,�8Pso�more�than�t��9w�o�1Qtriangles�cannot����Wt,��zbut�|�t��9w�o�|�t�nicely��:�.�RfIf�these�t�w�o�triangles�are�placed�\naturally"�then����Wb�A�oth�Bzof�Bytheir�w��9est�sides�will�lie�on��S��R�>�.���These�are�the�t��9w�o�Bzcongurations����Wdescrib�A�ed���in�part���(ii)�of�the�lemma.���The�last�t��9w�o���triangles�m�ust���b�A�e�placed����Win�b�this�conguration,�v;b�A�ecause�an��9y�other�placemen�t�w�ould�place�a�side�of����Wlength�t��b�t��along�a�side�of�length��c��in�at�least�t��9w�o�places,���eac�h�t�of�whic�h�t�w�ould����Wrequire�	$placing�t��9w�o�	$more�triangles�to�	#mak��9e�these�v�ertices�ha�v�e�t�yp�A�e�2���:�2,����Wcon��9tradiction.����d��src:1596 SevenTriangles.texThat�Tcompletes�the�pro�A�of�of�the�lemma.������R�color push gray 0��Lemma���7�	color pop������src:1600 SevenTriangles.tex�Supp��ose�#gthat�#fa��7�-tiling�c�ontains�a�#fnon-strict�vertex.�1�Then�the����Wtyp��e��of��that�vertex�is��2���:�1�,�$=the��tile�is�a�right�triangle�whose�smal�x�lest�angle����W�����is�����R�=�6����or���arcsin������P����1�����&�fe�����2�����"�v�,��and�3�triangles�me��et�at�the�non-strict�vertex.�X�(Se��e����WFigur��e�N<8.)�����>�color push gray 0�����22��Y�	color pop�����v�y�������>�color push gray 0�Y�	color pop���?������W�src:1605 SevenTriangles.tex�Pr��o�of�.���After��)the�previous�lemma,�]it�only�remains�to�rule��(out�cases�(ii)����Wand��(iii)�of�that�lemma's��conclusion.��W��:�e�rst�rule�out�case�(iii).��W��:�e�ma��9y����Wsupp�A�ose�&the�non-strict�%v��9ertex�is��V�8�,��the�midp�oin��9t�of�%the�maximal�segmen�t����W�P�H�Q�,��Zwith��Y�P��=�at�the�north,��Q��at�the�south;��\that�triangles��T�����1��*��,��[�T�����2���,�and��Y�T�����4�����W�are��Aon�the��Bw��9est�of��P�H�Q�,��sharing�v�ertex��V�8�,��and��Bthat��P�$�is�a�v�ertex��Bof��T�����1�����and����W�Q�P��is�P�a�v��9ertex�of��T�����2��*��,�_�and��S�ӯ�is�the�shared�w��9est�v�ertex�P�of��T�����1��{Q�and��T�����2��{R�and��R����W�is��Gthe��Hshared�w��9est�v�ertex��Gof��T�����4��#��and��T�����2��*��.�T��:�riangle��T�����3��#��is�east�of��P�H�Q�,���and�its����Weast���v��9ertex�is�called��E��2�.��J�P�H�V����=����V�8Q��=��S��R���=��a�.��IW��:�e���ha�v�e��R�>Q��parallel�to��S��V����W�and��T�S��P��8�parallel�to��R�>V�8�.��qBy�c��9hanging��U\north"�and�\south"�if�necessary��:�,����Ww��9e�
�can�assume�that�angle��R�>QV����=�������and�angle��S��P�H�V��=����
����.��There�are�four����Wcases�Tto�consider.�pNamely:����d��src:1616 SevenTriangles.texCase�T1:�p2�a����=��b�,�angle��E���=���x,�,�angle��V�8QE���=������d��src:1618 SevenTriangles.tex�Case�T2:�p2�a����=��b�,�angle��E���=���x,�,�and�angle��V�8QE���=��
����d��src:1620 SevenTriangles.tex�Case�T3:�p2�a����=��c�,�angle��E���=��
����,�angle��V�8QE���=�����,�and��b��6�=��c����d��src:1622 SevenTriangles.tex�Case�T4:�p2�a����=��c�,�angle��E���=��
����,�and�angle��V�8QE���=���x,�,�and��b��6�=��c����d��src:1624 SevenTriangles.tex�W��:�e���call�this���four-triangle�conguration��M����.�BAll�w��9e�ha�v�e�to���do�is�pro�v�e����Wthat�5`it�5_is�not�p�A�ossible�to�add�three�more�tiles�to��M�.C�and�thereb��9y�create�a����Wtriangle.�^This�+5could�b�A�e�+4done�b��9y�computer,�0�but�it�is�within�reac��9h�to�do�it����Wb��9y�Thand.����d��src:1628 SevenTriangles.tex�M�2��has�9�v��9e�exterior�9�v�ertices,�B�all�of�whic�h�ha�v�e�angles�9�less�than����_�(b�A�e-����Wcause��>they�are�comp�A�osed�of�t��9w�o��>angles�of�the��?tile�triangle,���not�b�oth��
����)����Wexcept�(�p�A�ossibly��P�q��in�case�1,�-�where�t��9w�o�(�angles��
��9�share�v��9ertex��P�H��.�VwIn�cases����Wwhere�Z�M�=�is�con��9v�ex�Zand�Yv�e-sided,�
Yplacing�three�Znew�triangles�m��9ust�lea�v�e����Wat�
Vleast�
Wt��9w�o�of�
Wthe�original�v��9e�edges�as�part�of�the�b�A�oundary�of�the�nal����Wtriangle.��Hence�j�t��9w�o�of�the�sides�of�the�nal�triangle�con�tain�sides�of��M����.����WThere�(�are�(�10�pairs�of�sides�of��M�!��to�consider;���in�eac��9h�case�w��9e�can�ask����Wwhether�dVit�dUis�p�A�ossible�to�dra��9w�a�third�side�and�ll�in�the�remaining�area����Wwith�Tcopies�of�the�tile.����d��src:1636 SevenTriangles.texCase���1�divides�in��9to���Case�1a�(when��
�T�=�����R�=�2),��Case�1b�(when��
�>����R�=�2),����Wand�

Case�1c�(when��
��.<�)��R�=�2).���Before�sub�A�dividing�in��9to�these�cases,�G7w�e����Wrst��argue�that��S��R����cannot�b�A�e��a�side�of�the�nal�triangle.��EAssume,��for����Wcon��9tradiction,�m�that�C�b�A�oth��E��2Q�C��and��S��R�T.�are�sides�of�the�nal�triangle.�֤Extend����W�E��2Q��t�and��u�S��R���to�their�in��9tersection�p�A�oin�t��L�.�
&The�nal�triangle��um�ust�include����Wtriangle��/�S��LE��2�,�ߞhence�m��9ust��0con�tain��/triangle��R�>QL�.�But�the�area�of�triangle����W�R�>QL��/�is��.v��9e�tiles,��not�three,�whic��9h��.w�e��/see�as�follo��9ws:��&Angle��R�>QL��3�=��2�
����,����Wsince�V\the�other�V[t��9w�o�V\angles�at��Q��are���X#�and���x,�.���Angle��S��R�>V����=����
����(b�A�ecause�it�is����Wopp�A�osite���S��V�%#�whic��9h��is�opp�ositiv��9e�angle��S��P�H�V�8�),�Pand�angle��V�R�>Q��l�=�����,�Psince����Wit�j�is�opp�A�osite�j��V�8Q� �=��a�.��S�j�Angle�j��L��=�����.��Since�the�angles�at�j��R�z��m��9ust�add����Wup�!Hto���R��,�$Eangle��QR�>L����=���x,�.�@MThen�!Hangle��QLR����=�������,�in�order�that�the�angles����Wof��9triangle��8�QLR�w�add�to���R��.��So�triangle��QLR�w�is�similar�to�the�tile,�/�and����Wsince�V�R�>Q����=��c�,�fQthe�similarit��9y�Vfactor�is��c=a�.���The�area�of��R�>QL��is�therefore����W�c���-=�2��*��=a���-=�2����times��othe��parea�of�a�single�tile.���T��:�o�complete�a�7-tiling�this�w��9a�y��ow�ould����Wth��9us���require�3�a���-=�2���{�=����c���-=�2��*��.�?�But��c���-=�2���=����a���-=�2���~�+����b���-=�2���=��a���-=�2���~�+���(2�a�)���-=�2���|�=�5�a���-=�2��*��,��^not���3�a���-=�2���.����W(Nev��9ertheless,��Git��Ddo�A�es�not�seem��Cthat�w�e�can�complete��Cthis�conguration�to����Wa���9-tiling,��but�that�is���irrelev��|ran��9t.)��This�con�tradiction�sho�ws���that�not�b�A�oth����W�E��2Q�b9�and��S��R�rx�are�sides�of�the�b:nal�triangle.�If��S��R�rw�is�a�side,�usthen�another����Wtriangle�t��T�����5���7�m��9ust�b�A�e�t�placed�east�of��E��2Q�.���If��T�����5���7�do�A�es�not�share�v��9ertices��E����and����W�Q�,���then��uat�least��tone�conca��9v�e�exterior��uv�ertex�is��tcreated,���whic�h�will��urequire����Wplacing����T�����6��ɦ�south�of����E��2Q��or�on��QE�"0�extended�north�of��E��2�.��W��:�e�will�then�ha��9v�e����Wv��9ertices�
[at�
Z�S���,���R�>�,��at��P�S=�unless��
�T�=�����R�=�2,�and�
Zat�one�of��Q��or��E��2�,��at�most�one�����>�color push gray 0�����23��Y�	color pop�������y�������>�color push gray 0�Y�	color pop���?������W�of���whic��9h�can�b�A�e�remo�v�ed���b�y�placing��T�����7��*��,��Dand�at�least���t�w�o�more�on�line��QE��2�,����Wwhic��9h��tis�to�A�o�man�y��:�.��Hence��T�����5���m�ust��sshare�v�ertices��E�i��and��Q�.��The�resulting����Wv��9e-triangle��sconguration��ris�con�v�ex.���Since�in�Case��r1,�Իangle��E�,�=�����x,�,�Ժthe����Wv��9ertex���at��E���remains���a�v�ertex.�muSince��E��2Q�M�=��c�,���angle����
�	8�do�A�es�not�o�ccur�in����W�T�����5��q��at�F��Q�,�SRso�the�v��9ertex�at��Q��remains�a�v�ertex�to�A�o.��8Then�the�v�e-triangle����Wcon��9v�ex��iconguration��jhas�six�v��9ertices.�	��T�����6���m�ust��ishare�the�existing�v��9ertices,����Was��if��w��9e�create�a�new�non-strict�v��9ertex�w�e�do��not�ha�v�e�enough��triangles�to����Wll���the���exterior�angles�th��9us�created�and�tile�a�triangle.���But�if��T�����6���z�shares����Wexisting��v��9ertices,�Jthe�resulting�six-triangle�conguration��will�b�A�e�con�v�ex,����Wand�`�will�ha��9v�e�`�at�`�least�v��9e�v�ertices,�s�so�cannot�b�A�e�completed�`�to�a�triangle����Wb��9y�Ʊadding�one�more�ưtriangle��T�����7��*��.�:This�con�tradiction�pro�v�es�ưthat��S��R����is�not����Wa�Tside�of�the�nal�triangle.����d��src:1662 SevenTriangles.texNo��9w�M_w�e�M^sub�A�divide�Case�1.�ĐFirst�consider�case�1a,��awhen��
�f�=�����R�=�2,����Wand���hence����S��P�H�E�U&�is�a�straigh��9t�line.�ROThis�is�the�only�case�in�whic��9h��M����is����Wa���quadrilateral,��rather�than�ha��9ving�v�e�sides.�ĦAs�pro�v�ed�ab�A�o�v�e,��w�e�will����Wha��9v�e�Zto�Zadd�one�triangle��T�����5�����w��9est�of��S��R�>�,��Jwith�w�esternmost�Zv�ertex��W�H��,����Wwith�bone�of�its�sides�con��9taining�segmen�t��S��R�rN�(whic�h�has�length��a�).��If�w�e����Wplace�Q�this�Q�triangle�so�that��W����lies�on��S��E���extended�and�on��R�>Q��extended,����Wthen��sw��9e��rwill�create�a�5-tiling.���Placing��T�����5����with�v��9ertices�at��S�2x�and��R����but����Wwith�2angle��
����at��R�BC�instead�of����2�will�create�a�conca��9v�e�2v�ertex�2at��R�BD�with�an����Wexterior�K�angle�greater�than�K���R�=�2,�Yaso�t��9w�o�K�tiles�w��9ould�b�A�e�required�at�v��9ertex����W�R�>�,�Qplea��9ving�Ejv�ertices�at��Q�,�Qo�E��2�,�Qp�S���,�and��W��M�at�Ekleast.���Placing�triangle��T�����5��p�with����Wone���side���along�line��S��R���but�extending�north�of��S�z��w��9ould�require�placing�at����Wleast��ht��9w�o�more��itriangles�north�of��S��E��2�,���lea�ving�v�ertices��i�R�>�,����Q�,��E��2�,�and��W�H��.����WPlacing��Otriangle��T�����5�����with�one�side�along��S��R���but�with�northernmost�v��9ertex����Won��*�S��R��h�south�of��S�/�w��9ould�require�us�to�place��T�����6�����on��S�R��i�north�of��T�����5��*��,���and����Wsince����S��R���=����a�,���T�����6�����w��9ould�extend�south�of����R�>�,�requiring��T�����7�����to�share�v��9ertex��R�>�.����WThen�tsev��9en�ttriangles�w�ould�tb�A�e�used�and�more�than�three�v��9ertices�w�ould����Wstill�T&exist,�c�e.g.����Q�,��E��2�,��W�H��,�and�the�T%south�v��9ertex�of��T�����6��*��.���Hence�the�triangle����Ww��9est�g�of�g��S��R�x�m�ust�b�A�e�g�placed�with��W����on��S��E���extended,�|vforming�a�5-tiling.����WW��:�e�Tare�then�ask��9ed�to�add�t�w�o�more�triangles�to�pro�A�duce�a�7-tiling.����d��src:1679 SevenTriangles.texSide�xx�W�H�E����has�length�xy2�b�PP�+��a�7�=�5�a�7>��2�c��=�2�����?�p����R���?�aH���u���5����UP�a�xy�and�xxhence�cannot�b�A�e����Wen��9tirely�ےco�v�ered�ۓb�y�placing�t�w�o�triangles�ۓnorth�of��W�H�E��2�.�	/Hence�no�triangle����Wcan��Pb�A�e��Qplaced�north�of��W�H�E��2�,���whic��9h�is�th��9us�a�side�of�the�nal�triangle.��oSide����W�W�H�Q�	��has�length�2�c��and�hence�if�one�triangle�is�placed�	�south�of��W�Q�,�Ga����Wsecond��Xone��Ym��9ust�b�A�e�placed�there�as�w��9ell;��if�these�are�placed�so�as�to�co��9v�er����Wall��of���W�H�Q��then�the�result�is�not�a�triangle.���Hence�no�triangles�can�b�A�e����Wplaced���south�of��W�H�Q�,��whic��9h�m�ust�b�A�e�a�second�side���of�the�nal�triangle.����WW��:�e��{are�th��9us��zask�ed��{to�place�t��9w�o��{triangles�east�of��E��2Q��and�complete�a�7-����Wtiling.��,If��these��t��9w�o�triangles��can�b�A�e�placed�so�that��Q��remains�a�v��9ertex����Wof���the�nal�triangle���then�the�v��9ertices�of�the�nal�triangle�will�b�A�e��W�H��,���Q�,����Wand��a�third�v��9ertex��U���east�of��E�p��on��W�H�E��extended.�?T��:�riangle��QE��2U���m��9ust�b�A�e����Wcomp�A�osed���of�t��9w�o���copies�of���the�tile��T�H��.��2These�t��9w�o���triangles�share�v��9ertices��E����W�and��another�v��9ertex��X��Q�on��QU����.�QThe�angle�at��X��R�is�a�righ�t�angle�b�y�Lemma����W1.��LAngle�S��P�H�E��2Q����=�����x,�.��KAngle��E�QX����=�angle��E�U���X���since�S�b�A�oth�are�opp�osite����Wside����E��2X��[�.�y�Hence���angle��QE�X�"�=�angle��U���E�X��[�,���and���these�angles�are�not��
����W�since���the���righ��9t�angles�o�A�ccur�at��X��[�.��QIf�they�are���. �then,��adding�the�three����Wangles�6�at��E��2�,�?w��9e�ha�v�e�3��A��=��z��R��;�Gwbut�since����A�=��arctan����(1�=�2),�?w��9e�do�not�ha�v�e����W��	��=�����R�=�3.�If��$angles��%�QE��2X���and��U���E�X���are�b�A�oth��$�����then�w��9e��%ha�v�e��$2��ˠ�+�����	��=�����R��,����Wbut��tthat��uis�imp�A�ossible�since����@�+��y��2��+��
�T�=�����>�and��t�
�>����.�{Hence��uit��tis�not�the�����>�color push gray 0�����24��Y�	color pop������y�������>�color push gray 0�Y�	color pop���?������W�case�Tthat��Q��is�a�v��9ertex�of�the�nal�triangle.����d��src:1696 SevenTriangles.texTherefore�}�w��9e�}�will�ha�v�e�to�}�extend�side��R�>Q��past��Q��b��9y�adding�another����Wtriangle�b�south�of��QE��2�.�uExtending�b�side��R�>Q��past��Q��will�require�creating�a����Wrigh��9t��angle��at��Q�;��if�that�is�done�with�one�triangle,�Othen�the�side�it�shares����Wwith�PJ�QE��}�will�ha��9v�e�PKlength�PJ�b�,�w�creating�a�non-strict�v��9ertex�at�distance��b��along����W�QE��2�,�T�whic��9h�H0has�H1length��c�.��That�will�create�a�conca��9v�e�v�ertex�H0with�exterior����Wangle�T�more�T�than���R�=�2,�d�whic��9h�cannot�b�A�e�lled�with�our�one�remaining�tri-����Wangle.�DaHence�"�b�A�oth�remaining�triangles�"�will�ha��9v�e�"�to�share�v��9ertex��Q�,�%�using����Wangles�������and����x,�.�SHence�the�shared�side�of�those�t��9w�o�triangles��cannot�ha��9v�e����Wlength����a��or��b��in���b�A�oth�triangles,���and�it�cannot�ha��9v�e�length����c��either�since�the����Wrst��one��has�its�side�of�length��c��along��QE��2�.��Hence�these�t��9w�o�new��triangles����Wdo��not��ev��9en�share�a�v��9ertex�along�their�shared�side,�Ehand�cannot�form�a����W7-tiling.�pThat�Tdisp�A�oses�of�case�1a.����d��src:1706 SevenTriangles.texNo��9w�˫w�e�˪tak�e�up�case�1b,��@when�˪�
�E:>����R�=�2.�?tThen�the�b�A�oundary�of��M����W�is�`�conca��9v�e�`�at��P�H��,�sWso�in�addition�to�adding�a�triangle��T�����5���0�w��9est�of��S��R�>�,�sWwith����Ww��9esternmost�bv�ertex��W�H��,�u=w�e�m�ust�add��T�����6�����north�bof��P�H��,�u>with�a�v��9ertex�at��P�H��.����WSupp�A�ose,��4for���pro�of�b��9y�con�tradiction,��4that��
�T�6�=���2��R�=�3,�in�whic��9h���case��T�����6���S�do�A�es����Wnot�ll�the� v��9ertex�at��P�H��,�
�or�that��
�T�=���2��R�=�3�but��T�����6��2��is�not�placed�with�angle����W�
�t
�at��}�P�H��.�(Then�w��9e��|m�ust��}also�add��T�����7��$�with�a�v��9ertex�at��P�H��.�(That�w��9ould�mean����Wthat�	�no�triangles�can�b�A�e�added�south�of��P�H�E����or�w��9est�of��W��,�F�so�that��W����W�and���Q���m��9ust�b�A�oth�b�e��v��9ertices�of�the�nal�triangle,���and��R��Z�m��9ust�not�b�A�e�a����Wv��9ertex,�Tand��hence��lies�on��W�H�Q�.�0Moreo��9v�er,�Tnothing��can�b�A�e�added�touc��9hing����W�QE��2�,��so���QE���m��9ust�lie�on�one�side��of�the�nal�triangle.��Therefore�the�third����Wv��9ertex�!�is�either��E����or�lies�northeast�of��E��on��QE��extended.�A�It�follo��9ws�that����Wtriangle��v�T�����7���m��9ust�ha�v�e��uside��a��along��P�H�E��2�,�~since�it�has�a�v��9ertex�at��P�*Y�and����Wcannot�(�extend�(�b�A�ey��9ond��E���along�line��P�H�E��2�,�-�and��P�E�5��=��h�a��is�(�the�shortest�side����Wof��the�tile.���E���cannot��b�A�e�a�v��9ertex�of�the�nal�triangle,�4since��W�\��and��Q��are����Wv��9ertices���and�triangle��T�����7���^�do�A�es���not�lie�inside�triangle��W�H�QE��2�,���since�it�extends����Wnorth�"�of�"��E���along�line��QE��2�.�E
Therefore�only�t��9w�o�"�tiles�meet�at��E��2�.�E
Hence�b��9y����WLemma�V�1,�g�
��a�=�����R�=�2.��But�in�Case�1b,��
��a>����R�=�2,�so�this�is�a�con��9tradiction.����WThis��7con��9tradiction��6pro�v�es�that��
�T�=���2��R�=�3�and��6�T�����6����is�placed�with�angle��
�r��at����W�P�H��.����d��src:1721 SevenTriangles.texT��:�riangle�"��T�����5��MR�has�to�b�A�e�placed�w��9est�of��S��R�>�,�S4since�w�e�pro�v�ed�ab�A�o�v�e�that��S��R����W�cannot��b�A�e��a�side�of�the�nal�triangle.�	Assume,��for�pro�A�of�b��9y�con�tradiction,����Wthat��it�is��placed�with�its��a��side�along��S��R�>�.��Consider�the�three�in��9terior����Wangles��at��R�>�.��They�are�angle��QR�V����=�������,�	�angle��V�8R�S���=����
����,�	�and�angle��W�H�R�S����W�whic��9h�P<migh�t�P;b�A�e��
����or���x,�,�w�but�not���R�since�angle��W�ک�=�������,�w�b�A�ecause�it�is�opp�A�osite����W�S��R�>�.�D
If��3angle��W�H�R�S�G?�=��:�
�P��then�a�conca��9v�e��4exterior��3angle�exists�at��R��q�that����Ww��9ould�/nha�v�e�to�b�A�e�lled�/mb�y�one�more�triangle��T�����7��*��,�5�lea�ving�four�v�ertices��W�H��,����W�S���,�T��E��2�,�and���Q��still�presen��9t.���S����w�ould��b�A�e�a�v��9ertex�since��the�angles�there����Ww��9ould�ub�A�e�t��<�from��T�����6��*��,�L���;�from��T�����1���,�L������from��T�����4���,�L�and������from��T�����5���,�L�and�their����Wsum�1
is�2��"z�+� �2��8&�=���2��s]���2�
�C��=�2��R�=�3��6�=����.�o�F��:�our�1
v��9ertices�1remaining�means�a����Wtriangle�Q�is�not�created.��Hence�angle��W�H�R�>S�y,�6�=��'�
����.�Hence�angle��W�H�R�>S�y+�=��'��x,�.����WThen�7��W�H�R�>Q�7��is�a�straigh��9t�line.��GConsider�the�angles�at��S���.��HThey�are������W�from���T�����6��*��,�������from��T�����1���,��
��)H�from��T�����4���,��and�this�time��
�4��from��T�����5���.���Their�sum�is����W2�����+��.���Y�+��-�
�T�=������+����p>����R��.�nHence��Oa��Nconca��9v�e�exterior��Nv�ertex�exists�at��N�S�ST�after����Wthe�G|placemen��9t�G{of��T�����5��r"�and��T�����6��*��.���The�exterior�angle�is������+�/��
����,�Tto�A�o�large�to�b�A�e����Wlled��'b��9y�the��&placemen�t�of�one�more�triangle��&�T�����7��*��.�aThis�con�tradiction�sho�ws����Wthat�T�T�����5��?��cannot�b�A�e�placed�with�its��a��side�along��S��R�>�.�����>�color push gray 0�����25��Y�	color pop����%s�y�������>�color push gray 0�Y�	color pop���?������d��src:1735 SevenTriangles.tex�Hence�U�triangle��T�����5���9�m��9ust�b�A�e�placed�w�est�U�of��S��R�e��in�suc�h�a�U�w�a�y�that�not����Wb�A�oth�i��S���and��R�y��are�v��9ertices.�(Supp�ose,�~�for�pro�of�i�b��9y�con�tradiction,�~�that��T�����5�����W�is���placed���so�that�it�do�A�es�not�ha��9v�e���a�v�ertex���at��R�>�.���It�m�ust���ha�v�e�t�w�o���v�ertices����Won��Sline��S��R�>�,��of�whic��9h�sa�y��U��6�is�the�southernmost.��W��:�e�then��Rha�v�e�a�conca�v�e����Wexterior�qMv��9ertex�either�qNat��R����(if��U�j1�is�south�of��R�>�)�or�at��U�j1�(if��U��is�qMnorth�of����W�R�>�).��P�T�����7��m��will�CIha��9v�e�to�b�A�e�placed�CJto�ll�this�exterior�conca�vit�y��:�.��PSince��Q��and����W�E�F��will�ïremain�ðv��9ertices�after�the�placemen��9t�of��T�����7��*��,��Fthe�third�v��9ertex�m�ust����Wb�A�e�,��W�H��;�8rhence�,��S����lies�on�line��W�H�E��2�.�b�That�implies�that��T�����7��We�has��S����for�a�v��9ertex����Wand�`�the�`�sum�of�the�angles�at��S���m��9ust�b�A�e���R��.��ZThose�angles�are���b��from��T�����6��*��,����W��
'�from�`�T�����1��*��,�H������from��T�����4���,�H�and�an�unkno��9wn�angle�from�a�T�����5���.���The�unkno��9wn����Wangle���m��9ust�b�A�e����n��a��2��c��������=�b��
��R������.��fThis���cannot�b�e��
����,���nor�can�it�b�e�����,����Wsince��2��Z�=���
��!�=��2��R�=�3�implies���[�=���R�=�3�whic��9h�in��turn�implies���y��=�0,�).a����Wcon��9tradiction.�=Hence��the�angle��of��T�����5��c�at��S�g��is���	��=����
�0��������.�Hence������+����	��=����
����;����Wbut�)�also����F�+�ƀ��֤�=�^y��)����
����,�n�whic��9h�con�tradicts��
���=�^y2��R�=�3.�Y�(So�w�e�do�not����Wev��9en���ha�v�e���to�analyze�the�imp�A�ossible�situation�near��R�>�.)���This�con��9tradiction����Wsho��9ws��0that��R��n�m�ust�b�A�e�a��1v�ertex�of��T�����5����and��S�d5�is�not�a�v�ertex.�Since��S��R���=����a����W�is�v�the�shortest�v�side�of�the�tile,��
triangle��T�����5���Z�extends�north�of��S����along��S��R�>�.����WThen��M�T�����7��#��m��9ust�b�A�e��Nplaced�with�a�v��9ertex�at��S�|S�north�of��S��E��2�.�There�will�then����Wb�A�e��v��9ertices���E��2�,����Q�,��W�H��,��and�a�v��9ertex��north�of��S��on�line��S��E��2�.���That�is�four����Wv��9ertices�Tat�least,�so�no�triangle�is�created.�pThat�disp�A�oses�of�Case�1b.����d��src:1751 SevenTriangles.texNo��9w�g'w�e�g(tak�e�up�case�g(1c,�{�when��
���<�'�R�=�2.��T��:�riangle��T�����5�����m�ust�g(b�A�e�placed����Ww��9est��of��S��R�>�.��Supp�A�ose,���for��pro�of��b�y�con�tradiction,���that�it��is�placed�with����Wangle����@�at��R��R�and�angle��
���at��S���.���Then�the�rst��v��9e�triangles�form�a����Wquadrilateral�V!with�V"v��9ertices�at��W�H��,�fU�Q�,�fT�E��2�,�and�V!�P��,�and�V!straigh��9t�angles�V"at��R����W�and�I��S���.��Consider�the�I�p�A�ossibilit��9y�that��W�H�S�P����is�I�a�side�of�the�nal�triangle.����WThen���triangle��T�����6���}�m��9ust���b�A�e�placed�with�a�side�along��P�H�E�M�and�a�v��9ertex�at��P�H��.����WBut��-the��,angle�at��P���will�then�b�A�e�at�least�2�
��W�+�n�����,��cwhic��9h�is�more�than���R��.����WHence�ۑ�W�H�S��P�$t�cannot�b�A�e�a�side�of�ېthe�nal�triangle.�	/But�the�length�of��W�P����W�is�+�2�b�,�1sand�w��9e�ha�v�e�+�2�b��F>��Ec��since�2�b��F�=��E4�a�>��3�a��=��a�8�+��b�>��Fc�.�_�Hence�+�at�+�least����Wt��9w�o���triangles�will���ha��9v�e���to�b�A�e�placed�north�of��W�H�S��P��.��lBut�that���will�lea��9v�e����Wv��9ertices��,�W�H��,����Q�,����E��2�,�and�another�v��9ertex�north�of��+�W�H�S��P��{more�than�three.����WThis���con��9tradiction�sho�ws�that��T�����5���n�cannot�b�A�e���placed�w�est�of��S��R���with�angle����W����at�f��R�w0�and�f�angle��
���at��S���.��PIf�it�is�instead�placed�with�angle��
���at��R�w1�and�angle����W���U�at��)�S���,��^there�will�b�A�e�a�conca��9v�e��(exterior��)angle�at��R�>�,�and�con��9v�ex��)v�ertices����Wat���W�H��,�;M�S���,�;L�P��,��E��2�,�;Land���Q�.��Ev��9en�if��it�is��p�A�ossible�to�ll�the�exterior�angle����Wat����R����with��T�����6��*��,���that�still�lea��9v�es���v�e�v�ertices�in�a�con�v�ex�conguration,���or����Wmore��Lthan��Mv��9e�v�ertices��Mwith�some�conca��9v�e��Lexterior�angles.�mIn�either�case,����Wa�7triangle�6cannot�b�A�e�created�b��9y�adding��T�����7��*��.�Hence��T�����5��,��cannot�b�A�e�placed�in����Weither�$�orien��9tation�$�with�its��a��side�along��S��R�>�.�KTherefore��T�����5��O��m��9ust�b�A�e�placed����Wwith��a�side��extending�past��S��R�>�,��either�north�or��S�u��or�south�of��R���along�line����W�S��R���(or��Yb�A�oth).�rSupp�ose,��&for�pro�of��Zb��9y�con�tradiction,��%that��T�����5����extends�north����Wof����S�h��to�a���v��9ertex��N�޷�on��R�>S�h��extended,��but�has��R���for�a�v��9ertex.���Then��T�����6�����W�m��9ust�(Xb�A�e�placed�(Wnorth�of��P�H�S��]�to�ll�the�conca��9v�e�(Xexterior�angle�at��S���.�U{That����Wlea��9v�es�icon�v�ex�iv�ertices�at�i�W�H��,�}��N����,�}��E��2�,�and��Q�i�at�least.��There�will�also�b�A�e�a����Wv��9ertex�:�at�:��R�J��unless��T�����5��eE�has�angle������there;�MDthat�m�ust�:�b�A�e�the�case�since�v��9e����Wcon��9v�ex�u�v�ertices�is�to�A�o�u�man�y�to�create�a�triangle�b�y�placing��T�����7��*��.�=Then��T�����5�����W�has�\/angle���]��at��N�U�and��
�߽�at�\.�W�H��.��Hence��N���S��4�has�\/length��c�=t��=u�a�.��T��:�riangle��T�����6�����W�m��9ust��Balso�eliminate�v�ertex��P�H��,��=or�there�will�again�b�A�e�v�e�v�ertices,��=whic�h����Wis��to�A�o�man��9y��:�.��lT�o��do�that,�?<�T�����6��.O�m�ust�ll�the�en�tire�angle��N���S��P�g��=���
����,�?=and����Wm��9ust�Umsupply�an�angle�Unequal�to�������8��2�
���at��P�H��.�ܻSince��S��P�E~�=����b�,�et�T�����6����has�angle�����>�color push gray 0�����26��Y�	color pop����;��y�������>�color push gray 0�Y�	color pop���?������W�����at�)��N����,�.�so�)�it�m��9ust�ha�v�e�)���+f�at��P�H��.�YRTherefore����`�=�����nj����2�
����.�Adding������+���
��-�to����Wb�A�oth�>sides�of�this�equation�w��9e�ha�v�e���(9�=�Տ��|�+�)U������)V�
����,�H+whic�h�=�implies���M��=�Տ�
����.����WW��:�e��Bno��9w��Aha�v�e�a�6-tiling��Aof�quadrilateral��N���E��2QW�H��.��9But�this�quadrilateral����Wis�Tda�parallelogram:����N���P�H�E�ז�is�parallel�to��W�R�>Q�Tc�b�A�ecause�transv��9ersal��P�Q����W�mak��9es��,equal��-alternate�in�terior��-angles��R�>QP�a�=�/��X�and��QP�H�E��a�=��
����,��"and����W�QE��is��Nparallel��Mto��W�H�N�}1�b�A�ecause�transv��9ersal��N���E���mak�es�complemen�tary����Wcorresp�A�onding��win��9terior��xangles��W�H�N���P�_9�=�V���l�+�����v��and��QE��2P��=�V�����=��
����.���By����Wplacing���one���more�triangle��T�����7��*��,��one�cannot�turn�a�parallelogram�in��9to�a����Wtriangle.��This�/con��9tradiction�sho�ws�that��T�����5��+��cannot�0b�A�e�placed�with��R�m�for�a����Wv��9ertex.����d��src:1783 SevenTriangles.texNo��9w���supp�A�ose,�փfor�pro�of�b��9y�con�tradiction,�փthat��T�����5��ڇ�is�placed�with�one����Wv��9ertex� �at� ��S����and�a�second�v��9ertex��U���south�of��R�15�on��S��R��extended.�?WThen� ��T�����6�����W�m��9ust��b�A�e�placed�south�of��R�>Q��to�ll�the�conca��9v�e��exterior�angle�at��R�>�.��That����Wlea��9v�es�z{con�v�ex�v�ertices�at��W�H��,����U����,����E��2�,�and��P��^�at�least.�K�There�will�also�b�A�e�a����Wv��9ertex�O�at��S����unless��T�����5��z��has�angle�O��
��}�there;�m<that�m�ust�b�A�e�the�O�case�since�v�e����Wcon��9v�ex�u�v�ertices�is�to�A�o�u�man�y�to�create�a�triangle�b�y�placing��T�����7��*��.�=Then��T�����5�����W�has�r�angle�r���tG�at��U�kd�and�����at��W�H��.��*Hence��U���R����has�length��b�Ȑ��ȑ�a����=��a�,��so�the�r�angle����Wof�(��T�����6��S,�opp�A�osite��U���R�8��m��9ust�b�e�(�����.�VT��:�riangle��T�����6��S-�m��9ust�also�(�eliminate�v�ertex�(��Q�,����Wor��Nthere��Owill�again�b�A�e�v��9e�v�ertices,�Mwhic�h��Ois�to�A�o�man�y��:�.��_T�o��Odo��Nthat,�M�T�����6�����W�m��9ust���ha�v�e���a�v�ertex���at��Q�,�׽and�m�ust�ll���the�en�tire���angle��U���R�>Q����=�����x,�,�׽and����Wm��9ust���supply���an�angle�equal�to�������a���c���a��
�����at��R�>�.���Since��R�Q�b��=��c�,��8�T�����6����has����Wangle��a�
�b��at��U����,��,so�it�m��9ust�ha�v�e����(�at��Q�.�
uTherefore�to�eliminate�v�ertex��Q��w�e����Wm��9ust��?ha�v�e��>2��X��+�V����y�=�GN��R��.�c0This�implies���I�=�GM�
����,��zso�the�tile�is�an�equilateral����Wtriangle.�T�No��9w�}�w�e�}�ha�v�e�a�6-tile�con�v�ex�}�conguration�with�v�ertices�}�at��W�H��,����W�U����,����E��2�,��P�H��,�and����S���.�-�Placing����T�����7���t�can�decrease�the�n��9um�b�A�er���of�v��9ertices�b�y�at����Wmost���one,���since�the���conguration�is�con��9v�ex.���Hence�no���nal�triangle�can����Wb�A�e���created.��sThis���con��9tradiction�sho�ws�that���triangle��T�����5���Q�cannot�b�A�e�placed����Wwith�Ta�v��9ertex�at��R�%��or�at��S���.����d��src:1799 SevenTriangles.texHence�Q`triangle��T�����5��|�is�placed�w��9est�of�Qa�S��R�>�,�`cwith�t�w�o�v�ertices�on�line��S��R����W�somewhere,�W�but�Snot�at��R�'��or�at��S���.�"mThen�at�least�t��9w�o�Sconca�v�e�exterior����Wv��9ertices��will��
b�A�e�created�somewhere�on�line��S��R�>�,��8whic��9h�m�ust�b�A�e��
lled�b�y����Wplacing��1triangles��0�T�����6�����and��T�����7���with�one�v��9ertex��1eac�h�on��0line��S��R��n�and�a�side����Wcon��9tained�ixin�iyline��S��R�>�.��That�will�create�t��9w�o�ixnew�v��9ertices�on�line��S��R�>�,�~�sa��9y����W�N��2�to��Othe��Nnorth�and��U��2�to�the�south.��_W��:�e�then�ha��9v�e�v�ertices��O�W�H��,�-L�E��2�,�-M�N����,����Wand�>j�U����,�H�ev��9en�if�straigh�t�angles�at��P��M�and��Q��are�created�b�y�placing��T�����6��i�and����W�T�����7��*��.��OF��:�our�]�v��9ertices�]�do�not�mak��9e�a�triangle,�o�so�this�placemen��9t�of��T�����5���F�is�also����Wcon��9tradictory��:�.�pThat�Tdisp�A�oses�of�case�1c,�and�with�it,�of�case�1.����d��src:1842 SevenTriangles.texNo��9w�g�w�e�g�tak�e�up�Case�g�2.��In�that�case��S��R�x�is�parallel�to��P�H�Q��since�the����Walternate�.�in��9terior�angles��S��V�8P�w��and�.��R�>S�V�<��are�.�b�A�oth�equal�to���x,�.�ϑW��:�e�ask�whic�h����Wpairs�¦of�sides�of�the�p�A�en��9tagon�§�S��P�H�E��2QR����could�b�e�(con��9tained�in)�sides�of����Wthe�Tnal�triangle.����d��src:1845 SevenTriangles.texCase��&2a:��Z�S��R��d�and��'�P�H�E�DX�are�sides.�aLet��N���b�A�e�the�in��9tersection�p�A�oin�t��'of��S��R����W�extended�&�and�&��E��2P�o��extended.�QKThen�triangle��N���S��P�o��has�angle�����at��S����and��
����W�at�/��P�H��,�6xb�A�ecause�the�angles�at��S����and��P�x��m��9ust�add�to���R��.�k�Then�angle��N�(��is�����,����Wand��triangle���N���S��P���is�similar�to�the�tile��T�H��.��But�it�has�side��b����=�2�a��opp�A�osite����Wangle������,�<�so�its�area��is�four�times�that�of�the�tile.��It�therefore�requires����Wfour��rtriangles��scongruen��9t�to��T�����1����to�co��9v�er�triangle��r�N���S��P�H��,��mwhic�h�is��seigh�t�total,����Wmore�Tthan�sev��9en.����d��src:1852 SevenTriangles.texCase�$h2b:�:��S��R�4��and��QE����are�$gsides�of�the�nal�triangle.�I�Then�w��9e�ha�v�e�to����Wadd���triangle����T�����5���|�south��9w�est�of����R�>Q�,���and�the�third�side�will�ha��9v�e�to���b�A�e�east�of�����>�color push gray 0�����27��Y�	color pop����TB�y�������>�color push gray 0�Y�	color pop���?������W�P�H�E����and�e�require�e�at�most�t��9w�o�triangles�e�to�complete�the�gure,���one�of�whic��9h,����Wsa��9y�&��T�����6��*��,�*�will�ha�v�e�&�to�b�A�e�north�of��S��P�H��,�*�sa��9y��S�P�H�N�k�and�one,�*�sa��9y�&��T�����7��*��,�*�will�ha�v�e����Wto���b�A�e���east�of��P�H�E��2�,��ysa��9y��P�E��2X��[�.��But���it�will�not�b�A�e�p�ossible���for��T�����7��*��,��xadding����Wjust�N,one�angle�N+at��E��2�,�\bto�ha��9v�e�N,a�side�extending��QE��2�,�\bsince�then�b��9y�Lemma����W1,��the��vtile��uw��9ould�con�tain�a��urigh�t�angle,��so��u�
�s��=����R�=�2,�and�triangle��v�P�H�E��2Q����W�w��9ould�K�ha�v�e�K�t�w�o�righ�t�angles,�Y^one�K�at��Q��and�one�at��E��2�.���This�con��9tradiction����Wdisp�A�oses�Tof�case�2b.����d��src:1861 SevenTriangles.texCase�z@2c:��H�S��R��~�and�zA�S�P��#�are�sides.�K4Since��P�H�E��r�and�zA�QE��are�not�sides�(b��9y����WCase��2a��and�Case�2b),��there�m��9ust�b�A�e�t�w�o��new�triangles�sharing�v��9ertex��E��2�,����Wone���on���eac��9h�side�of�the�existing�triangle��P�H�QE��2�.��>These�triangles�m��9ust�share����W(resp�A�ectiv��9ely)�V�side��P�H�E����and�side��QE��2�,�gsince�V�otherwise�additional�triangles����Wwill�pkha��9v�e�pjto�b�A�e�placed�sharing�v��9ertex��P��N�or��Q�,��0and�a�triangle�will�not�b�A�e����Wcreated.�9gTh��9us��w�e��ha�v�e�triangle���T�����5��̅�=����P�H�E��2F�g��and�triangle��T�����6��̄�=����QE��2G�,�!fwith����W�F�H�E��2G����and��R�>QG��and����S��P�F���straigh��9t�lines.�6�Then���angle��P�F�E�@$�=����
�Lc�(b�A�eing����Wopp�A�osite����P�H�E�3��whic��9h�has�length����c�)�and�angle��E��2P�F�ݎ�=����
�49�(so�that����S��P�F����is����Wstraigh��9t,�>�since�6Sangle��V�8P�H�S�K��=�������and�angle��QP�E�K��=�������),�>�so�angle��P�E��2F���=��������W�and��the��tile�is�isosceles�with�����=����
����,�ڄsince�triangle��P�H�E��2F����has�t��9w�o��angles����W�
����.��Angle�_�QE��2G�!��=��
����since�the�sum�`of�angles�at��E����is���R��.��Angle��G�!��=������W�since�PTit�is�PUopp�A�osite��QE�wM�=���a�.��pHence�angle��E��2QG���=����lH�=��
����.��qNo��9w�PTthere�are����Wthree�o�angles��
��W�at��Q��and�since��R�>QG��is�o�a�straigh��9t�angle,��e�
���=�(���R�=�3.�+�Hence����Wthe��*isosceles�tile�is��)actually�equilateral.���But�that�con��9tradicts�the�equation����W�b����=�2�a�.�pThat�Tdisp�A�oses�of�Case�2c.����d��src:1872 SevenTriangles.texNo��9w�MGsupp�A�ose,�[Dfor�pro�of�MHb��9y�con�tradiction,�[Dthat��S��R�]��is�MH(con�tained�in)�a����Wside�,�of�,�the�nal�triangle.�b�Then�since��S��P�u��is�not�an�edge,�2�w��9e�m�ust�,�place�a����Wtriangle,��sa��9y���T�����5��*��,��north�of�SP��:�,��and�since��P�H�E�n��is�not�an�edge,��w��9e�m�ust��place����W�T�����6�����east�y�of�y��P�H�E��2�,��%and�since��QE��-�is�not�an�edge,��$w��9e�m�ust�place�y��T�����7�����south�of����W�QE��2�.�Then��w��9e��coun�t�v�ertices:�c�W��:�e�ha�v�e��the�north�v��9ertex��N����of��T�����5����north����Wof����S��P�H��,��9and�the�south���v��9ertex�of��T�����7��*��,�south�of����QE��2�,��:and��R�>�.�p�In�addition,��:�S����W�will��5b�A�e�a�v��9ertex��6unless��N���lies�on��R�>S�:�extended;��&and��N���m�ust�also�b�A�e�the����Wthird�1�v��9ertex�triangle��T�����6��*��,�8�whic�h�has��P�H�E����for�one�side.�qConsider�the�angles����Wat�E��P�H��.��cAngle��QP�E�f
�=�����G��b�A�ecause�angle��P�QE�f�=����
�Ɉ�and�angle��E��-�is�opp�A�osite����W�P�H�Q�B��=��b�B��=�2�a�.�[JAngle���S��P�Q�B��=��
����.�[JAngle���S��P�N�x��m��9ust�b�A�e��
����,��.since�if�it��is������W�or��)��x,�,��1the�remaining��(angle��N���P�H�E�kZ�will�b�A�e���:��or�more,��1but�it�is�a�single�angle����Wof�.rtriangle��T�����6��*��.�g�Since�there�are�.qfour�angles�at��P�wU�and�one�of�them�is�����,�4�w��9e����Wha��9v�e���
�T>����R�=�2.��Since��in�triangle��N���S��P�H��,��X�
���is�used�at��P�H��,��Xangle��N���S��P���is�either����W���h�or�����x,�.��VThen�the���angles�at��S���are���x,�,�������,�and���angle��N���S��P�H��;��Gthe�total�is�at����Wmost�7���&��+�%-2�����whic��9h�is�less�than�7���R��,�@_since���CU<��*�=�2��)�<�
����.���Hence�7��N�0��do�A�es�not����Wlie�Φon�Χ�R�>S�Q��extended,���after�all.�HgThis�con��9tradiction�sho�ws�Χthat��S��R����is�not����Wcon��9tained�Tin�a�side�of�the�nal�triangle.����d��src:1886 SevenTriangles.texW��:�e��therefore��m��9ust�place��T�����5��3|�w��9est�of��S��R�>�.�ELet��W�Q��b�A�e�its�w��9estern�v�ertex.����WUnless����T�����5���6�is�placed�so�that�at�least���one�of�v��9ertices��S���and��R����are�eliminated,����Wi.e.�Z�W�H�S��P�r��is�)�a�)�straigh��9t�angle�or��W�H�R�>Q��is�a�straigh��9t�angle,�.�then�there�will����Wb�A�e���six�v��9ertices,��to�o�man��9y�to�allo�w�the���creation�of�a�triangle�b�y�placing����Wt��9w�o��3more�copies�of��T�����1��*��.��
Supp�A�ose,��+for�pro�of�b��9y�con�tradiction,��+that��W�H�S��P����W�is��!a�� straigh��9t�angle.���Angle��V�8S��P�é�=�z������and�angle��V�8S��R���=�z���x,�.���Hence�angle����W�W�H�S��R���=����
����.�~Since����S�R��=����a���b�A�ecause�it�is�opp�osite���to�angle��S��V�8R���=�������,���angle����W�S��W�H�R�‚�=��D����.�V�Hence�(�angle�(��W�R�>S�5I�=��C�����and�side�(��W�S�5I�=��D�b�.�V�W��:�e�will�(�sho��9w�that����Wno�eitriangle�ehcan�b�A�e�placed�north�of��W�H�S��P��.��If�ei�T�����6����is�placed�north�of��W�H�S��P����W�with��nnorth��mv��9ertex��N��Q�then�there�will�b�A�e�v��9ertices��Q�,��s�E��2�,��tand��N����.���Assume,����Wfor��pro�A�of�b��9y�con�tradiction,��
that��T�����6����has��W�H�P�:��for�a�side.��The�length�of�side�����>�color push gray 0�����28��Y�	color pop����k_�y�������>�color push gray 0�Y�	color pop���?������W�W�H�S��P�5�is��42�b�,�!�so�w��9e��3m�ust�ha�v�e��4�c����=���2�b�.��But��b��=�2�a�,�!�so��c��=�4�a�.��This�is����Wimp�A�ossible,���since���then���4�a����=��c�<�a�8��+�8��b��=�3�a�.���This�con��9tradiction���sho�ws�that����W�T�����6��nh�cannot�C�ha��9v�e��W�H�P����for�a�C�side.���Hence�t�w�o�triangles,�O\�T�����6��nh�and��T�����7��*��,�m��9ust�b�A�e����Wplaced�("north�(!of��W�H�S��P��.�T�If�("an��9y�side�of�length��a��is�placed�along��W�H�S��P�q�then����Ww��9e���need���at�least�three�triangles�(whic��9h�is�to�A�o�man��9y).��Supp�ose�triangles����W�T�����6��@��and���T�����7���are�placed��north�of��W�H�S��P�^��with�their��b��sides�along��W�H�S����and����W�W�H�P��.�g!Then�.9there�.:will�b�A�e�v��9ertices��Q��and��E��2�,�4sand�in�order�to�create�a�nal����Wtriangle,���there�m��9ust� b�A�e�straigh�t�angles� at��P���and�at��W�H��,���and�triangles��T�����6�����W�and��C�T�����7�����m��9ust�share��Ba�side.��;By�Lemma�1,���triangles��T�����6�����and��T�����7���ha��9v�e��Crigh�t����Wangles���at�the�shared�v��9ertex�on��W�H�S��P��.�s]Hence����
����=�PJ��R�=�2.�Then��T�����6���J�and��T�����7�����W�ha��9v�e��acute��angles�at��P�8��and��W�H��.��XHence�there�is�a�v��9ertex�at��W�H��,�&5as�w��9ell����Was���at����Q�,��,�E��2�,��-and�the�north�v��9ertex�of��T�����6��*��,��-so�no�triangle�is�created.�$�This����Wcon��9tradiction���sho�ws���that��W�H�S��P����is�not�a�straigh��9t�angle�after�the�placemen��9t����Wof�T�T�����5��*��.����d��src:1904 SevenTriangles.texIt�3wfollo��9ws�3vthat��W�H�R�>Q��is�a�straigh��9t�angle�after�the�placemen��9t�of��T�����5��*��.����WSince�Eangle�D�R�>QV����=�����}q�and�angle��QR�>V����=�������,�{w��9e�ha�v�e�angle�D�QV�8R���=����
����and����Whence�v��R�>Q�4�=�4�c�.�@xW��:�e�ha��9v�e�v�angle�v��V�8R�S���=�4�
��:�and�since�angle��W�H�R�Q�4�=�4���V�w��9e����Wha��9v�e���angle����W�H�R�>S�,�=����x,�.�*If�angle��W����=�������then��T�����5����has��S��R��O�=���a��for�a�side����Wand����W�H�S��P�8��is���a�striagh��9t�angle,��Pwhic�h���ha�v�e�already�dispro�v�ed.��Hence�angle����W�W����=���
����and�triangle��T�����5��F��has�side��c��along��R�>S���extended,��creating�a�conca��9v�e����Wexterior�@�v��9ertex�at��S���,�KYwhic�h�m�ust�b�A�e�lled�b�y��T�����6��*��.��Let�the�north�v�ertex�of����W�T�����5��a�b�A�e�6g�N����.��Then��N�S�K��=����a�,�>�so�6hit�is�p�A�ossible�to�place�triangle��T�����6����=����N���S��P�H��.��If����Wthis��5is�done,��nw��9e�ha�v�e��4a�six-triangle�con�v�ex�conguration��4with�v�e�v�ertices����W�W�H�QE��2P�N����.��(There�is�a�v��9ertex�at��P�^��since�the�angles�at��P�^��are�2��|�+���
��<����R��;����Wthere�Eis�Da�v��9ertex�at��N�(�since�the�angles�there�are������+�������<�AW�R��.)�%ASuc��9h�a����Wconguration�5�cannot�b�A�e�5�completed�to�a�triangle�b��9y�placing��T�����7��*��.�~gHence����Wtriangle����T�����6�����is�not����N���S��P�H��.�_But�an��9y�other�w��9a�y�of���placing��T�����6�����with�a�v��9ertex����Wat�;>�S��B�will�create�;=another�con��9v�ex�;>exterior�v�ertex�;>north�of�or�on��W�H�S��P��!�and����Weast���or���or�on��S��R����extended,���so��T�����7���5�will�ha��9v�e���to�b�A�e�placed�north�of��W�H�S��P����W�and���east�of����S��R���extended.�E�That�will�lea��9v�e���v�ertices�at����Q�,����E��2�,�and��W�H��,�as����Ww��9ell�"�as�at�least�one�v�ertex�north�of�"��S��P�H��,�&so�a�triangle�will�not�b�A�e�created.����WThis�Tcon��9tradiction�nally�disp�A�oses�of�Case�2.����d��src:1918 SevenTriangles.texW��:�e�s�no��9w�tak�e�up�Case�s�3.�7Since�the�angles�at��P��h�are��
���and���x,�,��the����Wp�A�ossibilit��9y�A�that��S��P�H�E��(�is�straigh�t�do�A�es�not�arise;�XG�M�:��is�a�con�v�ex�p�A�en�tagon.����WAdding�J�three�triangles�will�J�still�lea��9v�e�J�t�w�o�of�the�v�e�J�sides�on�the�b�A�oundary;����Whence��Sat��Tleast�t��9w�o�sides��Sof��M��7�are�(con�tained��Sin)�sides�of�the�nal�triangle.����WW��:�e��will�consider�eac��9h�of�the�ten��pairs�of�t�w�o�sides�and�sho�w�that�those����Wt��9w�o�Tsides�cannot�b�A�e�sides�of�the�nal�triangle.����d��src:1923 SevenTriangles.texCase���3a:����S��R��;�and����QE�/�are�sides�of�the�nal�triangle.�klThen�let��X����W�b�A�e�[their�[�in��9tersection�p�oin��9t.���The�transv�ersal�[��R�>Q��mak�es�[�alternate�in�terior����Wangles��;�R�>QX����=�J��
���and��QR�S�͡�=�J��
�۵�+�X'����with��QE�m�and��:�S��R��,���so��X�8��lies�to�the����Wsouth���of��R�>F�H��.��The���triangle��R�F�H�X���can�b�A�e�co��9v�ered���exactly�b��9y�four�copies�of����W�T�����1���&�(since��~it��is�similar�to�the�tile�but�has�side��c��
�=�2�a���opp�A�osite�angle�����),����Wbut��it��m��9ust�b�A�e�con�tained��in�the�nal�triangle,��Pwhic��9h�is�con�tradictory��:�,��Psince����Wonly�Tthree�more�triangles�are�a��9v��|railable.�pThat�disp�A�oses�of�Case�3a.����d��src:1930 SevenTriangles.texCase���3b:�H��S��R����and����P�H�E�.��are�sides�of�the�nal�triangle.�� Then�let��X�_��b�A�e����Wtheir���in��9tersection�p�A�oin�t;�9triangle��T�����5��#S�will�b�A�e�required�to�co�v�er����S��X��[P�H��.��Then����W�X��[P�H�E�F6�m��9ust��b�A�e��a�side�of�the�nal�triangle,��por�else�w��9e�will�need�to�place����W�T�����6���(�and����T�����7���north�of����X��[P�H�E��2�,���lea��9ving�v�ertices��R�>�,����Q�,���and���at�least�t��9w�o���north����Wof�tor�on�s�X��[P�H�E��2�,��con��9tradiction.�+�Therefore��X�P�H�E����is�a�side.�+�Since�s�E��2Q��is�not�����>�color push gray 0�����29��Y�	color pop�����}�y�������>�color push gray 0�Y�	color pop���?������W�a���side,��rw��9e�m�ust�place�triangle��T�����6���F�with���a�side�on�line��QE��2�.�QSupp�A�ose,��rfor����Wpro�A�of�	b��9y�con�tradiction,�Rvthat��T�����6��=��do�A�es�not�ha�v�e��E��;�for�a�v�ertex.��Then�a����Wcon��9v�ex� exterior�v��9ertex� is�created�on�line��QE��2�,�"�whic��9h�m�ust�b�A�e� lled�b�y��T�����7��*��.����WIf�ѻ�T�����6���c�extends�north�Ѽof��E�T��along��QE�T��then��T�����7���b�m��9ust�b�A�e�placed�with�a�side�on����W�X��[E��2�;�\�but�D�that�w��9ould�create�D�another�exterior�v�ertex�on��X��[E��2�,�P�b�A�ecause��X�E����W�has��Nlength��Ogreater�than��c�,���and�so�no�triangle�w��9ould�b�A�e�created.��If,���on�the����Wother�^Vhand,�p��T�����6�����has�its�north�^Uv��9ertex�on��QE���south�of��E��2�,�p�then��T�����7�����m��9ust�b�A�e����Wplaced���east�of����QE��2�,��Olea��9ving�parallel���sides��R�>Q��and����X��[E��,��Oso�no���triangle�is����Wcreated.�2_Hence�q��T�����6�����do�A�es�ha��9v�e�q��E��*�for�a�v��9ertex.�No�w�q�supp�A�ose,��"for�pro�of�b��9y����Wcon��9tradiction,��Gthat�r��T�����6�����do�A�es�not�r�ha�v�e��Q�r��for�a�v��9ertex.�5If�the�south�v��9ertex����Wof�d�T�����6�����is�dnorth�of��Q�,�w�then��T�����7���m��9ust�b�A�e�dplaced�east�of��QE��2�,�w�lea�ving�parallel����Wsides����R�>Q����and��X��[E��2�,� so�no�triangle�is�created.��If�the�south��9w�est���v�ertex��U��w�of����W�T�����6��`,�lies�5�on��QE����south�of��Q�,�=�then�since��QE�J��=��m�b��and��T�����6���has��E����for�a�v��9ertex,����W�T�����6����m��9ust��ha�v�e�side��c��along��QE��2�,��Hand��T�����7����will�ha�v�e�to�b�A�e�placed�south�of��R�>Q����W�and���w��9est���of��QE��2�,��?in�order�to�ll�the�conca��9v�e�exterior���angle�at��Q�.���That�will����Wlea��9v�e�T�v�ertices�at�T��X��[�,�d��R�>�,�d��U����,�and�the�southeast�v��9ertex�T�of��T�����6��*��,�so�no�triangle����Wis�ccreated.��This�ccon��9tradiction�pro�v�es�that�c�T�����6�����has��Q��for�a�v��9ertex,�vyas�w�ell����Was�T�E��2�.�pIn�other�w��9ords,��T�����6��?��has��QE����for�a�side.����d��src:1951 SevenTriangles.texLet��Y�%T�b�A�e�the�third�v��9ertex�of��T�����6��*��.�!�Since��QE�›�=�?i�b�,�W�angle��E��2Y�8Q��=���x,�.����WIf���angle��Y�8QE�Y(�=���
����,�xthen��E��2Y���is�parallel�to��X��[R�>�,�and�w��9e���ha�v�e���t�w�o�pairs����Wof��
parallel�sides�in�the�six-triangle��conguration,���a�problem�that�cannot����Wb�A�e��xed�b��9y�placing��T�����7��*��.���Hence�angle��Y�8QE�y?�=������.�Assume,� �for�pro�A�of�b��9y����Wcon��9tradiction,�buthat�S�
�֖�is�not�a�righ�t�angle.�ՍThen�w�e�ha�v�e�v�e�v�ertices��X��[�,����W�E��2�,��b�Y�8�,��c�Q�,�and��%�R�>�.��aIf��
�T<����R�=�2��&then�this�is�a�con��9v�ex�p�A�en�tagon,��band�cannot��%b�A�e����Wcompleted���to���a�triangle�b��9y�placing��T�����7��*��.��If��
�T>����R�=�2,���then�there�is�a�conca��9v�e����Wexterior��angle�at��E��2�,��whic��9h�p�A�ossibly�could��b�e�lled�b��9y��T�����7��<x�if��
�T�=���2��R�=�3,��but����Wthen��another��conca��9v�e�exterior��angle�w�ould�b�A�e��created�on��X��[P�H�E��1�somewhere,����Wsince�^the�length�^of��X��[P�H�E��5�is�more�than��c�.��}This�con��9tradiction�pro�v�es�that����W�
����is�(Ma�righ��9t�angle.�U[Therefore��P�H�E��2Y�6��is�a�straigh�t�angle.�U[No�w,�-ho�w�ev�er,�w�e����Wha��9v�e�,parallel�+sides��X��[P�H�E��2Y��d�and��R�>Q�;��since��X�P�H�E��2Y��d�has�length�+more�than����W�c�,�_�w��9e�P�m�ust�P�place��T�����7��{��south�of��R�>Q��with�its��c��side�along��R�>Q�.��Supp�A�ose,�_�for����Wpro�A�of�kb��9y�con�tradiction,��xthat��T�����7���&�has�angle�k���mF�at��R�>�.���Then�there�is�a�v��9ertex�at����W�R�>�,�߮since��Dthe�sum�of�the�angles�at��R���is�2�����+���
�T<����R��,�since�w��9e�kno�w��
�T�=�����R�=�2����Wand���hence,��pin�view�of����c����=�2�a�,��q�����=���R�=�6.���W��:�e���then�ha��9v�e�v�ertices���at��X��[�,��q�R�>�,��Y�8�,����Wand���the���third�v��9ertex�of��T�����7��*��,��~so�no�triangle�is�created.�tThis�con��9tradiction����Wpro��9v�es���that��T�����7��ʤ�do�A�es���not�ha��9v�e���angle������at��R�>�.��jHence�it�has�angle���)�at��R�>�,����Wmaking���a���straigh��9t�angle�there.���Then��T�����7���-�has�angle����N�at��Q�.���The�sum�of�the����Wangles��)at��Q��is�then���*�+���3�����=�����R�=�2�+�2��<����R��.�

Hence��)there�is�a�v��9ertex�at��Q�.����WSince�%`there�are�also�v��9ertices�at��X��[�,�)c�Y�8�,�and�%`the�third�v�ertex�of��T�����7��*��,�)cmaking����Wat�Tleast�four�v��9ertices,�no�triangle�is�formed.�pThat�disp�A�oses�of�Case�3b.����d��src:1978 SevenTriangles.texCase���3c:�u�S��R����and��R�>Q��are�b�A�oth���sides�of�the�nal�triangle.�!XSince�the����Win��9tersection�O�of�O�lines��P�H�S����and��QR�`�lies�w��9est�of��S��R�>�,�^w�P�H�S����is�not�a�side�of�the����Wnal�Edtriangle.���That�requires�the�placemen��9t�Ecof�a�triangle��T�����5��p
�north�of��P�H�S���.����WSince���P�H�E�w��is�parallel��to��R�>Q�,�,ew��9e�m�ust�place�a�triangle���T�����6��<�north�of��P�H�E��2�.����WThis�Ztriangle��T�����6�����cannot�reac��9h�all�the�w�a�y�to��S��R�jH�extended,�k9requiring�the����Wplacemen��9t��of���T�����7��-P�north�of��T�����5��*��.��pThis�cannot�lea��9v�e�a��triangle,�=�as�w�e��ha�v�e����Wv��9ertices�&at�&�Q�,�*>�R�>�,�and�a�&north�v��9ertex��X��j�on��R�>S���extended,�*>and�in�addition����Wa���fourth���v��9ertex�either�at��E��2�,���or�if��T�����6�����created�a�straigh��9t�angle�at��E��2�,���then����Wat��another�v��9ertex��of��T�����6��,,�on��QE����extended.��That�v�ertex�lies��east�of��QP����W�extended�Tand�hence�cannot�coincide�with��X��[�.�pThis�disp�A�oses�of�Case�3c.�����>�color push gray 0�����30��Y�	color pop�����	�y�������>�color push gray 0�Y�	color pop���?������d��src:1988 SevenTriangles.tex�Case��3d:���S��R���and���S�P�;��are�b�A�oth��sides�of�the�nal�triangle.��dThen�b��9y����WCase�j�3b�w��9e�w�ould�ha�v�e�j�to�place��T�����5���`�north�of��P�H�E��2�.��It�w�ould�ha�v�e�to�ha�v�e����Wangle�A���at��P�_#�in�order�@not�to�extend�north�of��P�H�S���.�5Then��E��r�w��9ould�b�A�ecome����Wa�ڧnon-strict�v��9ertex,��cwith��T�����5��N�extending�past��E��2�.��Since�b�y�Case�3c,��cw�e�m�ust����Wplace���one�triangle��T�����6���|�south���of��R�>Q�,���w��9e�m�ust���ll�the�conca�v�e�exterior�angle����Wat����E�{�with���a�single�triangle��T�����7��*��.��%Hence�b��9y�Lemma�1,�0��
����=�d��R�=�2.��T�����7��"��and����W�T�����5�����m��9ust�~
share�~a�common�east�v��9ertex��Y�8�,��8or�else�further�conca��9v�e�exterior����Wangles��are��formed.��A��9t��Y��D�the�sum�of�angles�is�2��	��=���2��R�=�3��<���,��so��there��are����Wv��9ertices�Tat��S���,�c��Y�8�,�and�Tthe�northeast�v�ertex��Z����of��T�����5��*��.�؀Since��Q��do�A�es�not�lie����Winside���triangle����S��Y�8Z����,�ѩthere�is�a�fourth�v��9ertex�and�no�triangle�is�formed.����WThis�Tdisp�A�oses�of�Case�3d.����d��src:2001 SevenTriangles.texCase�G�3e:��,�R�>Q��and��P�H�E����are�b�A�oth�sides�G�of�the�nal�triangle.���This�is����Wimp�A�ossible���since���they�are�parallel,��b�ecause�transv��9ersal��P�H�Q����mak�es�equal����Walternate�Tin��9terior�angles������with��R�>Q��and��P�H�E��2�.�pThat�disp�A�oses�of�Case�3e.����d��src:2004 SevenTriangles.texCase��!3f:��R�>Q��and��S��P���are�b�A�oth�sides�of��"the�nal�triangle.���The�in��9ter-����Wsection�!�p�A�oin��9t�of�lines�!��R�>Q��and��S��P�H��,�d�sa�y�!��W��,�d�lies�!�to�the�w��9est�of��S��P�j��and����Wtriangle���S��P�H�W���is��congruen��9t�to�the�tile.�.aSince�b�y�Cases�3a��to�3d,��#�S��R��8�is����Wnot��a��side�of�the�nal�triangle,�
�w��9e�m�ust��place��T�����5��6Z�as�triangle��S��P�H�W��.�;Since����W�P�H�E�J��is��|parallel��}to��R�>Q�,��w��9e�m�ust��}place��T�����6���$�north�of��P�H�E�J��and�south�of��S��P����W�extended.��PThis�T�is�only�T�p�A�ossible�if��T�����6��E�has��P����for�a�v��9ertex�and�has�angle������W�there,���so��the�north�side�of��T�����6����extends�segmen��9t��S��P�H��,���i.e.�there�is�a�straigh�t����Wangle��(at��P�H��.��That�creates�a�conca��9v�e��(exterior�angle�at��E��2�,���since��P�E���=����a�<�b����W�(w��9e�{�cannot�ha�v�e��a�<��=�<��b�{��since��c��=�<�2�a�).�O�Let��N�t��b�A�e�the�{�v��9ertex�of��T�����6���~�on��S��P����W�extended�|�and�|�let��X�14�b�A�e�the�south�v��9ertex�of��T�����6��*��.�R�W��:�e�m�ust�|�therefore�place����W�T�����7��@p�with��a�v��9ertex�at��E����and�a�side�along���QE��and�a�side�along��P�H�X��[�.��Unless����W�T�����7���r�is���triangle����QE��2X��[�,��more�exterior�conca��9v�e�angles���(and�hence�no�triangle)����Wwill�ɥb�A�e�formed;���hence��T�����7���L�is�triangle��QE��2X��[�.�6Then�considering�the�triangles����Wmeeting���at����E��2�,��Pb��9y�Lemma�1,��Q�
��O�=�q���R�=�2.���T��:�riangle��QE��2X�P�has�angle����~�at��Q����W�and����V'�at��X��[�.�vcThe��sum�of�the�angles�at��Q��is���)�+���2���<��1�R��,�$so�there�is�a����Wv��9ertex�[!at��Q�.���The�sum�of�the�angles�at��X�|�is�[ 2��~H<��R��,�l�so�there�is�a�v�ertex����Wat���X��[�.�There�are�also��v��9ertices�at��N����and��W�H��,���so�no�triangle�is�formed.�That����Wdisp�A�oses�Tof�Case�3f.����d��src:2019 SevenTriangles.texCase�˛3g:����R�>Q��and��E��2Q��are�b�A�oth�sides�of�the�˜nal�triangle.��By�Cases�3a����Wto�N�3d,�]�S��R�^��is�not�N�a�side�of�the�nal�triangle,�]so�w��9e�m�ust�N�place��T�����5��y^�w�est�of����W�S��R����but�p\touc��9hing�p]segmen�t��S��R�>�.�-�Assume,��for�pro�A�of�p]b�y�con�tradiction,��that����W�R�$B�is�not�a�v��9ertex�of��T�����5��*��.�Then�the�southern�v��9ertex�of��T�����5��>��on��S��R�$C�lies�north����Wof�LE�R�\��and�south�LFof��S���.��CBut��S�R����=��X�a�,�Zso�the�p�A�ortion�of��S�R�\��south�of��T�����5��v��has����Wlength��6less�than��a��5�and�cannot�b�A�e�co��9v�ered��6b�y�a�triangle��5lying�north�of��R�>Q����W�extended;�wthat��will�lea��9v�e��a�conca��9v�e��exterior��angle�that�cannot�b�A�e�lled.����WThis��3con��9tradiction�pro�v�es�that��2�R��q�is�a�v�ertex�of��T�����5��*��.��There�are�t�w�o�p�A�ossible����Worien��9tations��[of��Z�T�����5��*��:��~either�it�has�its��a��side�along��S��R�>�,��so�that��W� =�lies�on����W�S��P�Xm�extended,�Nor��it�has��its��c��side�along��S�R�>�,�Nso�that���S����is�the�midp�A�oin��9t����Wof�ovthe�oueast�side�of��T�����5��*��.�*�Assume,���for�pro�A�of�b��9y�con�tradiction,���that�ov�S��z�is�the����Wmidp�A�oin��9t���of�the���east�side�of��T�����5��*��.��
Let��N����b�e�the�north���v��9ertex�of��T�����5��*��,�@and����W�W��N�its�;kw��9est�v�ertex.���Then��W�H�N�4O�is�parallel�to��QE��2�,�D�since�the�corresp�A�onding����Win��9terior�n�angles�made�n�b�y�transv�ersal��W�H�Q��are�angle�n��N���W�Q�&u�=��
���and�n�angle����W�E��2QR���=�����|��+�z���x,�,�t	whic��9h�K�are�supplemen�tary��:�.��;Since��E��2Q�K��is�(in�Case�3g)�a�side�of����Wthe�~�nal�~�triangle,����N���W��f�cannot�b�A�e,���so�triangle��T�����6���+�m��9ust�b�A�e�placed�north��9w�est����Wof����N���W�H��.��Then��T�����7���{�has�to���ha��9v�e���a�v�ertex���at��S�A��to�ll�the�conca��9v�e���exterior����Wangle��8there.��But�b��9y�Case�3e,��q�P�H�E�1k�is�not�a�side,�so�some��9triangle�has�to�����>�color push gray 0�����31��Y�	color pop���� �0�y�������>�color push gray 0�Y�	color pop���?������W�b�A�e�߷placed�north�of��P�H�E��2�;��but�߶w��9e�ha�v�e�no�more�triangles,��pso�a�con�tradiction����Whas�Tb�A�een�reac��9hed.�pThat�disp�oses�of�Case�3g.����d��src:2035 SevenTriangles.texCase��3h:����S��P�-z�and��QE�g��are�b�A�oth�sides�of��the�nal�triangle.��:W��:�e�note����Wthat���the�in��9tersection���p�A�oin�t����X�]C�of�those�t��9w�o���sides�lies�to�the�northeast�of����W�P�H�E��2�,��^b�A�ecause���of���the�alternate�in��9terior�angles�made�b��9y�the�transv��9ersal��P�H�Q�.����WHence���triangle��T�����6���~�will�ha��9v�e���to�b�A�e���placed�with�its��a��side�along��P�H�E��2�.���But����Wthen�dFits�dG��f
�angle�is�not�at��P��)�and�it�cannot�lie�south�of��S��P��)�extended,����Wcon��9tradiction.�pThat�Tdisp�A�oses�of�Case�3h.����d��src:2040 SevenTriangles.texCase�%F3i:�<U�S��P�n)�and��P�H�E��x�are�b�A�oth�sides�%Gof�the�nal�triangle.�LGBy�Case�3e,����W�R�>Q�V�is�Vnot�a�side�of�the�nal�triangle,�f.and�b��9y�Case�3h,�f.�QE��5�is�not�a�side.����WBy�YPCases�YO3a�to�3d,�jO�S��R�i��is�not�a�side.��cTherefore�w��9e�will�ha��9v�e�YPto�place��T�����5�����W�w��9est�>of�=�S��R�>�,�x�T�����6��D��south�of��R�>Q�,�xand��T�����7��D��southeast�of��QE��2�.�+-�T�����7���will�ha��9v�e�=to�>b�A�e����Wplaced��zwith�a��{v��9ertex�at��E�?��in�order�to�a��9v�oid��zcreating�a�conca��9v�e��zexterior����Wv��9ertex��on��QE��2�.�I�Let��W���b�A�e��the�w�est�v�ertex�of��T�����5��*��;�+�this�m�ust��b�A�e�the�w�est����Wv��9ertex�/�of�/�the�nal�triangle.�lIt�m�ust�therefore�/�lie�on��S��P�x��extended.�lHence����W�T�����5����has����W�H�S�>��for�its�north�side.�	If��T�����5���has��R���for�a���v��9ertex,��}then�there���is�a����Wconca��9v�e�mexterior�angle�lat�the�w��9estern�v�ertex�lof��T�����6����on��R�>Q��or��R�>Q��extended.����WHence�a��T�����5���P�has�its�southern�v��9ertex��Y�o��south�a�of��R�q��on��S��R��extended.�o�T�����5���P�has����Wangle���
�s��at��S���,��vsince�the�sum�of�angles��there�m��9ust�b�A�e���R��,�and�angle���h+�at��W�H��,����Wsince�.ethe�.dside�opp�A�osite��W�wH�is�greater�than��S��R����=����a�.�g�Hence��T�����5��Y�has�angle������W�at���Y�8�.�n�But��W�H�R��Q�is�parallel�to��P�E��2�,��Cso�the�in��9tersection��p�A�oin�t��of�lines��W�Y����W�and�͕�P�H�E�P��will�lie�north�of��S��P��.��Hence��S��P��,����P�E��2�,���and��W�Y����cannot�b�A�e�sides�of����Wa�(triangle�including�an��9y�p�A�oin�ts�(suc�h�as��V�8�)�south�of��S��P�H��.�T�But�in�Case�3i,����Wb��9y�vih�yp�A�othesis�vj�S��P��L�and��P�H�E����are�sides�of�the�nal�triangle,���and��W�H�Y����m��9ust����Wb�A�e�g[a�side�gZsince�it�is�on�the�b�A�oundary�of�the�sev��9en-triangle�conguration.����WThis�Tcon��9tradiction�disp�A�oses�of�Case�3i.����d��src:2053 SevenTriangles.texCase�r�3j:�׳�P�H�E��'�and��QE��(�are�b�A�oth�sides�of�r�the�nal�triangle,��]then�as�w��9e����Wha��9v�e��already�sho��9wn,���none�of��S��P�H��,��S�R�>�,�and���R�Q��can�b�A�e��sides.�7bHence�w��9e����Wwill�!�ha��9v�e�!�to�place�the�remaining�three�triangles�on�those�sides,�%
sa��9y��T�����5��L��on����W�S��R�>�,�LR�T�����6��k��on�AS�R�Q�,�and�AS�T�����7��k��on�AR�S��P�H��.��l�T�����6���m��9ust�b�A�e�placed�ARwith�its��c��side�on��R�>Q����W�(or���else���a�conca��9v�e���exterior�angle�will�b�A�e�created),��so�its�angle�at��Q��will����Wnot��b�A�e��
����,��and�the�sev��9en-triangle��conguration�has�a�v�ertex�at��Q�,��since����Wthe��7angle�sum��6there�is�less�than���R��.��Assume,��pfor�pro�A�of�b��9y�con�tradiction,����Wthat���T�����5�����and��T�����7���do�not�ha��9v�e��the��same�third�v��9ertex.�U]Then�w�e��ha�v�e�four����Wv��9ertices{those�L
t�w�o�Lplus��E��?�and��Q�.���That�con��9tradiction�sho�ws�that�L�T�����5��v��and����W�T�����7���^�m��9ust���share���their�third�v�ertex,��Psa�y��N����.�ӚLet��X�\�b�A�e���the�southern�v�ertex����Wof�{��T�����6��*��.�O�F��:�or�a�triangle�to�b�A�e�formed,��_the�sides�m��9ust�b�e��N���P�H�E��2�,��_�X��[QE��,�and����W�N���R�>X��[�.���No��9w�D`�R�S�c4�=��/�a��so�in��T�����5��*��,�P#�N���S��e�m��9ust�b�A�e��b��or��c�.���In��T�����7���,�P#�S��P�)�=��/�b��so��N���S����W�m��9ust�F�b�A�e��a��or��c�.���Therefore��N���S�g'�=��!�c�,�Sand�angles��N�R�>S����and��N�P�H�S����are�b�A�oth����W�
����.��The��angle��	sum�at��P����is�then�2�
��?�+�v����=�����R��.��But�since���xx�+�v��
��?�+����=�����R��,����Wthis��'yields��(�
�&^�=�������,��so�the�tile�is�equilateral,��con��9tradicting�2�a����=��c�.��That����Wcon��9tradiction�Tdisp�A�oses�of�Case�3j.����d��src:2065 SevenTriangles.texW��:�e���ha��9v�e�no�w���sho�wn�that�no�pair�of���sides�of��M����can�b�A�e�sides�of�the�nal����Wtriangle.�pThis�Tcompletes�Case�3.����d��src:2067 SevenTriangles.texNo��9w�Tw�e�tak�e�up�Case�4.�pAs�in�Case�3,��M�8�is�con�v�ex.����d��src:2122 SevenTriangles.texW��:�e��rst�pro��9v�e��that��the�three�sides�of�the�nal�triangle�cannot�b�A�e��S��P�H��,����W�P�H�E��2�,�dand�!)�R�>Q�.�?�Supp�A�ose,�for�pro�A�of�b��9y�con�tradiction,�dthat�those�are�the����Wthree��~sides.���The�in��9tersection�p�A�oin�t��W��a�of�lines��S��P��`�and��R�>Q��lies�to�the�w�est����Wof��G�S��R�>�,��b�A�ecause��Fthe�transv��9ersal��S�R����mak��9es�corresp�A�onding��Fin�terior�angles����W�R�>S��P���=������+�w�����and��O�S�R�Q���=��
���+�w�����,���and��Oin�Case�4�w��9e�ha�v�e��c��>�b�,���whic�h�����>�color push gray 0�����32��Y�	color pop����!ʡ�y�������>�color push gray 0�Y�	color pop���?������W�implies��j�
�T>����x,�.�"T��:�riangle��k�W�H�S��P�M�has�angle���;��at��R�Ө�(since�the�angle�sum�at��R����W�is�	/��R��)�and�angle��
����at��S��3�(since�the�angle�sum�at��S��4�is����);�
;hence�it�has�angle����W��7��at�5��W�H��,�=�opp�A�osite�side��S��R���=����a�.�}�Hence�triangle��W�S��P�~��is�5�congruen��9t�to�the����Wtile��}and��~w��9e�can�call�it��T�����5��*��.��~Let��X�]��b�A�e�the�in��9tersection�p�A�oin�t��}of�lines��P�H�E�,��and����W�R�>Q�,��whic��9h���exists�b�A�ecause�w�e�ha�v�e�assumed�the�third���side�is��R�>Q�.��~Then����Wtriangle�T��QE��2X���is�2-tiled�b��9y�triangles��T�����6��@�and�T��T�����7��*��.��?Hence�the�tile�is�a�righ�t����Wtriangle,�Q��
�e��=��U��R�=�2.��rSince�E�there�is�a�E�straigh��9t�angle�at��Q�,�Q�w�e�ha�v�e���Z��=��U��R�=�3����Wand�#hence������=�����R�=�6.�
Consider�the�angle�"sum�of�triangle��W�H�P�X��[�:��the�#angle����Wat��^�W��A�is�����,�Ԡthat�at��P��is��
����+�t?����,�Ԡso�that�at��X�b��is��������
������2�����=�����R�=�6���=�����.����WHence��;triangle��QE��2X�D��is�congruen��9t��:to�the�tile;�ͯit�cannot�b�A�e�2-tiled.��%This����Wcon��9tradiction��=pro�v�es��>that�the�three�sides�of�the�nal�triangle�cannot�b�A�e����W�S��P�H��,�T�P�E��2�,�and��R�>Q�.����d��src:2139 SevenTriangles.texW��:�e��shall��no��9w�argue�that�none�of�the�ten�pairs�of�sides�of��M����can�b�A�e����W(con��9tained�Tin)�sides�of�the�nal�triangle.����d��src:2142 SevenTriangles.texCase���4a:�#'�S��P���and��P�H�E���are�b�A�oth���sides�of�the�nal�triangle.���Then��QE����W�do�A�es�VPnot�VQlie�on�the�third�side,�|�b�A�ecause�of�the�alternate�in��9terior�angles�made����Wb��9y��@transv�ersal��A�P�H�Q��to��S��P�=#�and��P�H�E��2�.�jHence�triangle��T�����5����is�required�south�of����W�QE��2�.���T�����5�����m��9ust�w6b�A�e�placed�w5along��QE��h�with�a�v�ertex�w5at��E��2�,���since��P�H�E��h�is�a�side�of����Wthe��nal��triangle.��Let��X�|Z�b�A�e�the�east�v��9ertex�of��T�����5��*��.���S��R��=�cannot�lie�on�a�side����Wof���the�nal�triangle,���since�the�in��9tersection�p�A�oin�t�of��S��R���and��P�H�E�z��lies�north����Wof��!�S��P�H��,���b�A�ecause��"the�transv��9ersal��S�P���of��R�>S�'�and��P�H�E�T�mak��9es�corresp�A�onding����Win��9terior��angles��
�R�>S��P�ک�=�����9��+��������and��S�P�H�E���=����
�E6�+�������,���whose��sum�is���S�+�������>����R��.����WHence�)�triangle��T�����6��T��m��9ust�b�A�e�placed�w�est�of��S��R�>�.�ZULet��W�r��b�A�e�the�w�est�v�ertex����Wof�Hx�T�����6��*��.���Since��R�>Q��cannot�Hwb�A�e�a�side�of�the�nal�triangle�when��S��P��[�and��P�H�E����W�are�`Wsides�`X(as�sho��9wn�ab�A�o�v�e),�s�T�����7�����m�ust�b�A�e�placed�south�`Xof��R�>Q�.��zNo�w,�swhat����Wcan���b�A�e���the�v��9ertices�of�the�nal�triangle?���P��o�is�one�of�them;��zthe�others�m��9ust����Wb�A�e�b2�X���(the�east�b1v��9ertex�of��T�����5��*��),�uiand��W���(the�w�est�v�ertex�of��T�����6��*��).�	Then��S��7�is����Wnot�,�a�,�v��9ertex;�8ahence�triangle��T�����6��WY�has�angle��
��?�at��S���.�b�Since��R�>Q��is�not�a�side,����W�W��o�lies�\�to�\�the�w��9est�of�the�in��9tersection�p�A�oin�t��Y�j��of�\�lines��S��R�l��and��R�>Q�.��But����W�Y�8S��!�=�D�b��T�since��Uangle��Y�R�>S��!�=�D��x,�.�]qHence��U�W�H�S��=��c�.�]qThen��Tthe��U�
���angle�of��T�����6�����W�m��9ust�<b�A�e�=opp�osite��W�H�S���;�$1but�w��9e�=already�pro�v�ed�=�T�����6��I��has�angle��
����at��S���.�:)Since����Win�]�Case�4,�pw��9e�ha�v�e��b�
��6�=�
��c�,�pw�e�ha�v�e�����<�
�
��~�so�]��T�����6�����has�only�one�angle�equal����Wto�T�
����.�pThis�con��9tradiction�disp�A�oses�of�Case�4a.����d��src:2162 SevenTriangles.texCase�A�4b:�u'�S��P����and��QE����are�sides�of�the�A�nal�triangle.���The�in��9tersection����Wp�A�oin��9t�sl�X�'��of�these�t�w�o�sksides�lies�to�the�east,���b�A�ecause�of�the�alternate�in��9terior����Wangles�xmade�b��9y�the�wtransv�ersal��P�H�Q�.�'Supp�A�ose,�=for�pro�of�b��9y�con�tradiction,����Wthat��.�
����is��-a�righ��9t�angle.��Since�2�a�1�=��c��w��9e��.ha�v�e�����=�1��R�=�6�and����]�=���R�=�3.����WConsidering�p�the�angle�sum�of�triangle��P�H�QX��[�,���w��9e�ha�v�e���M�+�KD����at��P����and������W�at���Q�,���so��w��9e�ha�v�e���i���������2��	��=��������at���X��[�.��Hence��triangle��P�H�X�E�rJ�is�similar��to����Wthe��tile,��but��with���o�opp�A�osite�its��b��side��P�H�E��2�.��Hence�the�similarit��9y�factor�is����W�b=a��,�=��sin���(��R�=�3)�=�����sin���(��=�6)��+=����,���?�p���
a~���?�aH���u���3����|.�K�Hence�%+the�area�%*of�triangle��P�H�X��[E��\�is�three����Wtimes��that�of�the�tile,�Oand�after�it�is�tiled�there�will�b�A�e�no�more�triangles,����Wbut�$9there�will�still�b�A�e�v��9ertices�at��X��[�,�g��S���,��R�>�,�and�$9�Q�.�IThis�con�tradiction����Wsho��9ws�Tthat��
����is�not�a�righ�t�angle.����d��src:2173 SevenTriangles.texLet����W�6��b�A�e�the�in��9tersection�p�oin��9t���of�lines��R�>Q��and��S��P�H��.��,Supp�ose,�$
for����Wpro�A�of��b��9y��con�tradiction,��that��R�>Q���is�the�third�side.�5LThen�triangle��W�H�S��R�-��is����Wcongruen��9t���to���the�tile,��Dha�ving�angle���
��at��R�>�,��Dangle��
�N�at��S���,��Eand���angle������at��R��.����WGiv��9e���triangle��W�H�S��R��<�the�name��T�����5��*��.��Then�triangle��X��[P�E�0�is�2-tiled�b��9y��T�����6�����and����W�T�����7��*��;��Lhence��Hthe��Itile�is�a�righ��9t�triangle,��whic�h�w�e��Hha�v�e�sho�wn�is��Inot�the�case.����WThis���con��9tradiction���sho�ws�that��R�>Q����is�not�the�third�side.�ƘSince�triangle�����>�color push gray 0�����33��Y�	color pop����"Ạy�������>�color push gray 0�Y�	color pop���?������W�W�H�S��R�WZ�is�Gcongruen��9t�Gto�the�tile,�S�but��R�>Q��is�not�the�third�side,�S�at�least�t��9w�o����Wtriangles�FVwill�b�A�e�FUrequired�w��9est�of�line��S��R�>�,�R�and�at�least�one�south�of��R�>Q�.����WThat�will�use�sev��9en�triangles,��and�lea�v�e�none�to�tile�triangle��P�H�E��2X��[�.�That����Wdisp�A�oses�Tof�case�4b.����d��src:2184 SevenTriangles.texCase�;4c:�d�S��P�X�and��R�>Q�<�are�sides�of�the�nal�triangle.�hThese�lines�in��9ter-����Wsect���in���the�w��9est�v�ertex��W�H��.��.T��:�riangle��W�S��R����is�congruen��9t���to�the�tile,���and�w�e����Wcall��it��T�����5��*��.��^Then,��*b��9y�cases�4a�and�4b,��)triangles��T�����6���,��*and��T�����7�����will�ha��9v�e��to�ha��9v�e����Wsides���P�H�E�=L�and���QE��2�,��Lresp�A�ectiv��9ely��:�.�
�(If�these�triangles�do�not�ha��9v�e��v�ertices����Wmatc��9hing��Falready�existing�v�ertices,�no�triangle�will�b�A�e�formed.)�kLet��X����b�e����Wthe��_third�v��9ertex�of��T�����7��*��,�ԑand��N��C�the�third�v�ertex�of��T�����6��*��.�tThen�the�nal�trian-����Wgle�,bis�,c�W�H�X��[N����.�a�A�,\straigh��9t�angle�is�formed�at��P�H��,�2&and�since�angle��S��P�H�V��k�=��3�
����W�and���angle����QP�H�E�Z��=�׸����,�	�w��9e�m�ust���ha�v�e�angle����E��2P�H�N�М�=�׸��x,�.�g%But��P�E�Z��=�׹�b��so����Wthe�mWangle�at��N�f;�m��9ust�mXalso�b�A�e���x,�.�$yBut�in�Case�4,��Xw�e�ha�v�e��b�$v�6�=��c�mW�so������6�=�$v�
����.����WHence�%���%z�=��N����,�)�since��T�����6��P��has�t��9w�o�%�angles�equal�%�to���x,�.�M�Then�angle��P�H�E��2N��2�=��
����.����WNo��9w�~�supp�A�ose,��for�pro�of�b��9y�con�tradiction,��that�~�angle��QE��2X�F!�=��������=���x,�.��QThen����Wconsidering�Tthe�angle�sum�at��E����w��9e�ha�v�e�2�
����+�8�����=���R��.�pBut���������.2�
����+�8�������\�=������9 2(��`���8�������x,�)�+�����������\�=������9 2��`���8�3�������W�src:2197 SevenTriangles.tex�whic��9h���implies�����&��=�%��R�=�3.��-But�if������and����are�b�A�oth���R�=�3�then�so�is��
����,����Wcon��9tradiction,�O@since�x�b�4V�6�=�4W�c�.�
�This�con�tradiction�wpro�v�es�that�wangle��QE��2X����W�is��dnot��c��x,�.��Therefore�it�is��
����,��'and�the�angle�sum�at��E�/��tells�us�3�
��=�����R��,��'or����W�
�T�=�����R�=�3.�9But�үagain�Үthat�implies������=�����	��=��
����,��con��9tradiction.�9That�үdisp�A�oses����Wof�TCase�4c.����d��src:2203 SevenTriangles.texCase���4d:���S��P�B��and��S�R�
�are���sides�of�the�nal�triangle.�?Then�b��9y�Case�4a,����Wtriangle����T�����5���]�will���b�A�e�required�east�of��P�H�E��2�;���b��9y�Case�4b,���triangle��T�����6���]�will�b�A�e����Wrequired��south��of��QE��2�;��and�b��9y�Case�4c,��triangle��T�����7��D`�will�b�A�e�required�south����Wof���R�>Q�.��The�three��v��9ertices�of�the�nal�triangle�m��9ust�b�A�e��S���,��a�p�oin��9t���X��%�east����Wof����P���on��S��P��extended,��kwhic��9h�m�ust���b�A�e�the�east�v��9ertex�of��T�����5��*��,��kand�a�p�A�oin��9t��Y����W�south�of��R�Q�on��S��R�P�extended,��whic��9h�m�ust�b�A�e�the�south�v�ertex�of��T�����7��*��.�Since����Wtriangle��G�S��X��[Y���m��9ust��Hinclude��T�����6��*��,��Jlo�A�cated�south�of��QE��2�,��Ktriangles��T�����7���,��J�T�����6���,�and����W�T�����5����m��9ust��all��share�a�v��9ertex�on�line��Y�8X��[�,�׊with�an�angle�sum�of�����there.��But����Wthen�יthe�side�ךof��T�����7��@�along��R�>Q��m��9ust�b�A�e�longer�than��R�>Q����=��c�,���con��9tradiction.����WThat�Tdisp�A�oses�of�Case�4d.����d��src:2215 SevenTriangles.texCase�4e:���S��R�]�and��P�H�E��P�are�sides�of�the�nal�triangle.���Let��N���b�A�e�the����Win��9tersection��of��P�H�E����and���S��R�>�,�whic�h�lies�to�the�north�of��S��P�H��.��Then�triangle����W�N���S��P��h�is���similar���to�triangle��T�����1��*��,���since�it�has�angle�����at��P��g�and�angle��
��at����W�S��]�(and�KXhence�KY��M�at��N����),���but�the�side�opp�A�osite�angle���M�is��b�,���not��a�.��}So����Wtriangle��l�N���S��P�O�could�b�A�e�tiled�b��9y�an�in�tegral��mn�um�b�A�er��K�_T�of�copies�of��T�����1����only����Wif�+��K�[R�=��k(�b=a�)���-=�2��V��is�an�in��9teger.�`1If��K�[S�=��j3�then�a�triangle�is�not�formed,�1�since����Ww��9e���ha�v�e�v�ertices���at��N����,����E��2�,����Q�,�and��R�>�.��wIf��K�@�=�nX2�then��N���S��P���is���2-tiled,�so����W�
��P�=�"���R�=�2.�!iSince�lR�c��=�2�a��in�Case�lQ4,��w��9e�ha�v�e�lR��$��=�"���R�=�6�and������=���R�=�3.�!iThen����W�b=a�V��=��V�sin��<���x,=�����sin��p���X��=���Vş��?�p������?�aH���u���3�����,��so����K����=�V�(�b=a�)���-=�2���k�=�3,��not���2.�Hence��K����6�=�V�2.�But����Wthese�Tare�all�the�p�A�ossibilities�for��K����.�pThat�disp�oses�of�Case�4e.����d��src:2225 SevenTriangles.texCase�M�4f:��;�P�H�E����and�M��QE��are�sides�M�of�the�nal�triangle.�šThen�w��9e�require����W�T�����5���
�south�ncof��QR�~��b��9y�Case�4c,����T�����6���north�of��S��P��F�b��9y�Case�4a,���and��T�����7���w��9est�of����W�S��R�"��b��9y��Case��4e.�UIn�order�that�a�triangle�b�A�e�formed,�Q�the�new�triangles����Wm��9ust���share���existing�v�ertices���along�the�sides�men��9tioned,���and�w�e���m�ust�ha�v�e����Wstraigh��9t�_angles�_at��Q��and��P�H��.���The�v��9ertices�of�the�nal�triangle�are��E��2�,�q�the�����>�color push gray 0�����34��Y�	color pop����#���y�������>�color push gray 0�Y�	color pop���?������W�north��v��9ertex��N��~�of��T�����6��*��,��Zon���P�H�E�v��extended,�and��the�south�v��9ertex��X����of��T�����5��*��,��Zon����W�E��2Q���extended.�)�Then�the�three��new�triangles�m��9ust�share�a�v��9ertex��W�b��w�est����Wof���S��R�"$�and�the�angle��sum�there�m��9ust�b�A�e���R��.�LBut�then�the�north�side�of��T�����5�����W�will���b�A�e��W�H�Q�,���whic��9h�is�longer�than��c�,�since��R�>Q����=��c�.�	�That���con��9tradiction����Wdisp�A�oses�Tof�Case�4f.����d��src:2235 SevenTriangles.texCase���4g:�b��P�H�E�;��and��R�>Q��are�sides�of���the�nal�triangle.�-Then�w��9e�m�ust����Wplace����T�����5��"~�south���of��QE�{	�with�its��a��side�along��QE��2�.��In�case��
����=�L��R�=�2�this����Wcan��b�A�e�done,���with�the�third�v��9ertex��X��j�at��the�in�tersection�p�A�oin�t�of��P�H�E�bA�and����W�R�>Q�.��
Otherwise�l$more�l#triangles�will�b�A�e�required�south�of��QE��2�.��Since�neither����W�S��P���nor�{8�S�R��w�is�a�side�when��P�H�E��k�is,���b��9y�Case�4a�and�Case�{94d,�triangles��T�����6�����W�and�H��T�����7��s'�will�Hb�A�e�required�north�of��S��P��c�and�w��9est�of��S��R�>�,�UKso�there�can�b�A�e�no����Wsecond��Xtriangle�south�of��W�QE��2�.�Hence��
�T�=�����R�=�2.�Since��c����=�2�a�,���w��9e��Xthen�ha�v�e����W�����=�����R�=�6��and����	��=���=�3.��The��three�v��9ertices��of�the�nal�triangle�are��X��[�,��and����Wthe��pnorth�v��9ertex��q�N��T�of��T�����6��*��,���whic�h�lies��qon��P�H�E�)��extended,���and�the�w��9est�v�ertex����W�W���of�֟�T�����7��*��,��*whic��9h�lies�on��QR����extended�and�is�also�a�v�ertex�of��T�����6��*��.��If��T�����6��F�do�A�es����Wnot���ha��9v�e�v�ertices����W���and��P��or����T�����7�����do�A�es�not�ha��9v�e�v�ertices����S�W��and��R�>�,��no�nal����Wtriangle��Jis��Iformed.��QW��:�e�ha��9v�e��S��P����=�^��b�;���angle��J�R�>S�V�l��=�^���x,�;���angle��W�H�R�>S���=�^������W�b�A�ecause�b�the�angle�sum�at��R�s�is���R��;��hence��W�H�S����=���b�.��Hence��W�P��,�v"the�south����Wside��~of���T�����6��*��,��Bis�2�b�.��Hence��c����=�2�b�.��But��c��=�2�a�.��Hence��a��=��b��and�������=���x,�.��This����Wcon��9tradicts��C���r�=�����R�=�6��Dand�����=���R�=�3.�>This��Dcon��9tradiction�disp�A�oses�of�Case����W4g.����d��src:2250 SevenTriangles.texCase���4h:�Am�S��R���and��QE�+�are�sides�of���the�nal�triangle.���Because��P�H�Q��is����Wparallel�G�to�G��S��R�>�,�T�the�in��9tersection�p�A�oin�t�G��X��Z�of��QE��1�and�G��S��R�X<�lies�to�G�the�south����Wof����R�>�.���A��9t���least�one�triangle��T�����5���.�will�ha��9v�e���to�b�A�e�placed�inside�triangle��R�>QX��[�.����WSince��5�S��R��s�is��6a�side,����P�H�E�g�and��S�P���are�not�sides,���b��9y�Case�4e��6and�Case�4d,����Wso��triangles��T�����6��.:�and���T�����7���are��required�north�of��S��P�Lw�and�northeast�of��P�H�E��2�,����Wareas���whic��9h���cannot�in�tersect�triangle����R�>QX��[�.��VHence�triangle��T�����5���G�m�ust�b�A�e����Wexactly���triangle��R�>QX��[�.��7Angle��R�X��[Q����=�����b�A�ecause���angle����P�H�QE���=����and����P�H�Q����W�is���parallel���to��S��R�>X��[�.�But�angle��R�>X��[Q��is�opp�A�osite�side��R�>Q��{�=��c�,��\so���angle����W�R�>QX�F!�=����
����.�NHence�����	��=��
��.�OHence����b��=��c�.�NBut�in�Case�4�w��9e�ha�v�e��b����6�=��c�.�NThis����Wcon��9tradiction�Tdisp�A�oses�of�Case�4h.����d��src:2260 SevenTriangles.texCase���4i:��>Supp�A�ose��QE�&!�and��R�>Q��are�sides�of�the�nal�triangle.��NSince��QE����W�is��Ua�side,���none�of��P�H�E��2�,��S��P��,�and��U�S��R����are�sides,�b��9y�Case�4f,�Case�4b,�and����WCase�!�4h,�$�so�triangles�!��T�����5��*��,�$��T�����6���,�and�!��T�����7��Lp�m��9ust�b�A�e�!�placed�on�those�three�sides,����Wresp�A�ectiv��9ely��:�.���Let�9��X��&�b�e�the�east�v��9ertex�of��T�����5��*��,�B�and�9�let��Y�H�b�e�the�north��9w�est����Wv��9ertex�e�of��T�����6��*��.�
Then�these�m�ust�b�A�e�the�v�ertices�of�the�nal�e�triangle,�y�so��Y����W�is��`also��_the�w��9est�v�ertex��_of��T�����7��*��,��^and��T�����7���is�triangle��S��R�>Y�8�,��^�T�����6���is�triangle��P�H�S��Y�8�,����Wand����T�����5���i�is�triangle����P�H�E��2X��[�.���Since�only���t��9w�o���triangles�meet�at��E��2�,��zb��9y�Lemma�1����W�
�)��=��]��R�=�2.�A~Then�!�b�A�ecause��c��=��\2�a��w��9e�ha�v�e�!����#�=��\��R�=�6�and�����=��\��R�=�3.�ABut�at��S���,����Wthere�_�are�_�four�angles,�rXt��9w�o�_�of�whic��9h�are���a��and���x,�,�rWso�the�angles�of��T�����6���d�and����W�T�����7��$�at��X�S�|^�m��9ust�add��Yto�3��R�=�2,���whic�h�is��Yimp�A�ossible.�That�disp�oses��Yof�Case�4i.����d��src:2269 SevenTriangles.texCase��'4j:���R�>Q��&�and��S��R��d�are�sides�of�the�nal�triangle.�h�By�Case�4c,����WCase�y4g,��and�yCase�4i,�neither��S��P�H��,��P�E��2�,��
nor�y�QE��K�can�b�A�e�ysides�along�with����W�R�>Q�.���Therefore�C�w��9e�m�ust�place�C�triangles��T�����5��*��,�O[�T�����6���,�and�C��T�����7��ng�along�those�sides,����Wresp�A�ectiv��9ely��:�.���Let����X�YS�b�e�the�southeast���v��9ertex�of��T�����7��*��,��qand��N����the�north�v�ertex����Wof����T�����5��*��.���Then��X��[�,����N����,�and����R���m��9ust�b�A�e�the���v�ertices�of���the�nal�triangle.���Hence����W�X��1�lies�?�on��R�>Q��extended�and��N�8��lies�on�?��R�S����extended.���The�v��9ertex��Y�N�of��T�����6�����W�that�Wdo�A�es�not�Wlie�on��P�H�E��N�extended�m��9ust�lie�on��N���X��[�.���Then��Y�eS�m��9ust�lie�on����W�S��P���extended,���so�ɫthat�ɬ�S�P�H�Y����is�the�south�ɫside�of��T�����5��*��;���and��Y��m��9ust�ɫlie�on��QE����W�extended,�Սso�śthat��QE��2Y����is�the�north�side�of��T�����7��*��.��That�is,�lines��S��P�~�and��QE�����>�color push gray 0�����35��Y�	color pop����$h�y�������>�color push gray 0�Y�	color pop���?������W�meet�XDat�XC�Y�8�.��?Since�only�t��9w�o�XCtriangles�lie�to�the�north�of��E��u�and��QE��2Y�f{�is�a����Wstraigh��9t���line,��qb�y���Lemma�1�w�e�ha�v�e����
�T�=�����R�=�2.��Then�b�A�ecause��c��=�2�a����w��9e�ha�v�e����W��#�=�!I��R�=�6�koand����t�=���R�=�3.��Because�the�angle�sum�at��P��R�of�the�angles��S��P�H�V�8�,����W�QP�H�E��2�,��#and���E�P�Y��O�is����R��,��#w��9e�m�ust�ha�v�e�angle��E��2P�H�Y����=�����x,�.��Because�the�sum�of����Wangles��2�QE��2P�>�and��Y�8E�P�>�m��9ust�b�A�e���R��,���w�e��1ha�v�e�angle��Y�8E��2P�ک�=����
����.��Hence�angle����W�E��2Y�8P�ک�=�������.�nBut��Lside��M�P�H�E���=��b��since��Lit�is�opp�A�osite�angle��P�H�QE��2�.�mHence�angle����W�E��2Y�8P�S�=�
/����=���x,�,�/�whic��9h��-is��,a�con�tradiction�since��,����=�
/��R�=�6�and����[�=���R�=�3.����WThat�Tdisp�A�oses�of�Case�4j.����d��src:2284 SevenTriangles.texThat�*completes�*all�ten�sub-cases�of�Case�4,�/?and�with�them,�/>the�pro�A�of����Wthat�Tcase�(iii)�of�the�previous�lemma's�conclusion�is�imp�A�ossible.����d��src:2287 SevenTriangles.texW��:�e�� no��9w�tak�e��!up�sho�wing�that�case��!(ii)�of�the�previous�lemma's�conclu-����Wsion��is��imp�A�ossible.��In�that�case�w��9e�start�with�a�v��9e-triangle�conguration����W�M����,�l�and�B�w��9e�m�ust�sho�w�it�is�not�p�A�ossible�to�B�mak�e�it�in�to�a�triangle�b�y�adding����Wt��9w�o��amore�copies�of��T�H��.�Since��S��P�6D�is�parallel�to��R�>Q�,��^those�t��9w�o��asides�cannot����Wb�A�oth���b�e�sides���of�the�nal�triangle.��Supp�ose,��vfor���pro�of�b��9y�con�tradiction,����Wthat�$�triangle��T�����6��OW�is�placed�$�south�of��R�>Q�.�J�Let��X���b�A�e�the�south�v��9ertex�of��T�����6��*��.����WThen�D�w��9e�D�ha�v�e�v�ertices�at�D��S���,�P��E��2�,�and��X��[�.���Supp�A�ose,�for�pro�A�of�D�b��9y�con�tradic-����Wtion,��Zthat�v&�R�>Q�v%�is�not�a�side�of��T�����6��*��.�>�Then�w��9e�will�ha�v�e�v%to�place��T�����7�����with�a����Wv��9ertex���on��R�>Q�.��mThen�w�e�still�ha�v�e���v�ertices�at��X��[�,�¨�S���,�and����E��2�,�so�the�nal����Wtriangle���m��9ust���b�A�e��S��X��[E��2�.��Then��T�����6��~�and��T�����7���m��9ust�2-tile�triangle��R�>QX��[�.��Then����Wangle����QR�>X��,�is�less�than�a���righ��9t�angle,�1so�angle��W�H�R�>X��-�is�not�a�straigh��9t����Wangle,��7since���angle��W�H�R�>Q��is�a�righ��9t�angle.�s^This�con�tradiction�sho�ws�that����W�R�>Q�(J�is�(Ia�side�of��T�����6��*��.�UPSince��R�>Q��_�=��b�,�-�T�����6��R��has�(Ja�righ��9t�angle�at��R�8��and���*�at��Q�,����Wor�jWthe�other�jXw��9a�y�jWaround.�zUnless�the�righ��9t�angle�of��T�����6�����is�at��R�z��and��S��P�H�E����W�is�m�straigh��9t,���the�resulting�six-triangle�conguration�will�m�ha�v�e�v�e�or�more����Wcon��9v�ex�Ggv�ertices,�S�and�cannot�Ghb�A�ecome�a�triangle�b�y�placing�Ghone�more�cop�y����Wof����T�H��.��Therefore����T�����6�����m��9ust�b�A�e�placed�with�its�righ�t�angle�at��R�>�.��Let��X�E9�b�A�e����Wits���south�v��9ertex.�ٹThen��S��X�^�is�longer���than��c��since��S�R��e�=��(2�a��=��c�.�ٹHence����Ww��9e��cannot��create�a�triangle�b��9y�placing��T�����7��<��w�est�of��S��R�>�.�NAlso��S�P�H�E��2�,��ev��9en�if����Wit�o�is�o�a�straigh��9t�line,��has�length��b�JW�+�JX�a�(>�c�,��so�o�w��9e�cannot�create�a�triangle����Wb��9y�r'placing��T�����7�����north�of��S��P�H�E��2�.�2�The�only�remaining�p�A�ossibilities�are�south����Wof��V�T�����6����or��Ueast�of��QE��2�.��vIn�either�case�the�triangle��T�����7����will�share�v��9ertex��Q�.����WThe�vangle�valready�at��Q��is���R�=�2�N�+�2����,��Bso�vto�mak��9e�a�triangle�w��9e�m�ust�vadd����W��R�=�2�mD���2����.��'If����S��P�H�E�'�is�not���straigh��9t,�NJw�e���cannot�p�A�ossibly�create�a�triangle;����Whence�t��S��P�H�E���is�straigh��9t.��Then������=����arctan������P����1�����&�fe�����2�����#��,��not�t���R�=�6,�so�t���=�2��c���d�2��v��is�neither����W����nor��K��x,�,��and�it��Lis�not�p�A�ossible�to�eliminate�the�v��9ertex�at��Q��b��9y�placing����W�T�����7��*��.���This�?con��9tradiction�?sho�ws�that�?�T�����6��i��cannot�b�A�e�placed�south�of��R�>Q��and����Wcompleted�Tto�a�triangle.����d��src:2314 SevenTriangles.texTherefore�YA�R�>Q��will�b�A�e�one�of�the�sides�of�the�nal�triangle.��7Then��S��P����W�cannot���b�A�e���a�side�of�the�nal�triangle,���since�it�is�parallel�to��R�>Q�.�7FHence����Ww��9e�7�m�ust�7�place��T�����6��b4�north�of��S��P�H��.��W��:�e�m�ust�not�7�create�a�conca��9v�e�v�ertex����Wan��9ywhere�5on�5�S��P�}��b�y�placing��T�����6��*��,�=
so�the�v�ertices�5of��T�����6��_��m�ust�include�5�S��!�and����W�P�H��,�Punless���S��P�E����is�straigh��9t�and��S��E��is�one�side�of��T�����6��*��;��but��S��E���=��k�b�z�+��a�>�c�,����Wso�1�that�1�is�not�p�A�ossible.�q�Hence��S��P�z��is�a�side�of��T�����6��*��.�q�If��S��P�H�E����is�straigh��9t,�8�i.e.����W����=���Farctan������P��  4�1��  4��&�fe�����2�����$��,��tthen��;w��9e�ha�v�e�created��:a�conca�v�e�v�ertex�at��:�P�H��,��uso�w�e��;m�ust����Wha��9v�e������=�����R�=�6.�In�that�case�the�total�angle�at��P�M��after�placing�the�����angle����Wof�%��T�����6��P8�there�is�%���R�=�2�+�����+���%�=������,�)�so�%�if�%��N�u�is�the�north�v��9ertex�of��T�����6��P8�w��9e�ha�v�e����W�N���P�H�E��Q�straigh��9t.�Q�W��:�e�'cannot�mak�e�a�triangle�b�y�placing��T�����7��Q��w�est�of��N���R�>�,�+�or����Wnorth��of��N���P�H�E��2�,���since�these�segmen��9ts�are��longer�than��c�,���so�that�lea��9v�es��east����Wof��0�QE�Eb�as��/the�only�p�A�ossibilit��9y��:�,��fsince�w�e�kno�w��/�R�>Q��m�ust�b�A�e�a��/side�of�the�����>�color push gray 0�����36��Y�	color pop����%(��y�������>�color push gray 0�Y�	color pop���?������W�nal��Otriangle.��bSince��QE���=�ji�b�,���w��9e�m�ust�place�the��b��side��Pof��T�����7�����along��QE��2�,����Wwith�.�the�righ��9t�angle�at��E��2�,�5Tb�A�ecause�the�sides�of�the�nal�triangle�m�ust�b�A�e����W�N���S��R�>�,��N�P�H�E����(extended),�and���R�>Q���(extended).��The�angle�of��T�����7��/t�at��Q��m��9ust����Wthen��b�A�e������.�5nThe�total�angle�at��Q��is�then���R�=�2��+�2���o�=���5��=�6,��not��enough��to����Weliminate�+]the�v��9ertex�at��Q�.�^�That�+^completes�the�pro�A�of�that�case�(ii)�of�the����Wprevious�_�lemma's�conclusion�_�is�imp�A�ossible,�rjand�that�completes�the�pro�A�of����Wof�Tthe�lemma.���􍍍�R�color push gray 0��Lemma���8�	color pop������src:2330 SevenTriangles.tex�A�N<7-tiling�c��annot�c�ontain�mor�e�than�one�non-strict�vertex.����W�src:2333 SevenTriangles.texPr��o�of�.�-dThe��previous��lemmas�ha��9v�e�sho�wn��that�eac�h��non-strict�v�ertex��is�of����Wt��9yp�A�e�Z�2���:�1�Z�and�o�A�ccurs�in�a�certain�3-triangle�conguration�(sho��9wn�in�Figure����W8.)��kSupp�A�ose��Fa�7-tiling�con��9tains�t�w�o��E(or�more)�non-strict�v�ertices.��kW��:�e�ha�v�e����Wto�=	consider�the�=
follo��9wing�cases:�k�Case�1,�F�the�t��9w�o�=	3-triangle�congurations����Wo��9v�erlap���(share�a�triangle),��th��9us�requiring�v�e�or�few�er�triangles;��@Case�2,����Wthe���t��9w�o���3-triangle�congurations�do�not�share�a�triangle,���but�share�a�side;����WCase��3,�νthe�t��9w�o��3-triangle�congurations�do�not�share�a�triangle�or�a�side.��z�ˍ�>�����color push gray 0�mM}���9��src:2341 SevenTriangles.tex����J�A�ٖ̲W����>,����P����>,�D�Q����.I�ٖ�V����5����Y����5�ٖ�X����u*���8E��U[��k" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.9226 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.9226 42.67912 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.9226 85.35825 L 0 setlinecap stroke  end �o" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 0.0 moveto 73.9226 85.35825 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 0.0 moveto 147.84457 42.67912 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 0.0 moveto 110.88391 64.01868 L 0 setlinecap stroke  end �v" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 85.35825 moveto 147.84457 42.67912 L 0 setlinecap stroke  end �v" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 85.35825 moveto 147.84457 85.35825 L 0 setlinecap stroke  end �x" tx@Dict begin STP newpath 0.8 SLW 0  setgray  147.84457 42.67912 moveto 147.84457 85.35825 L 0 setlinecap stroke  end �������J��color push gray 0Figure�UU10:�q�Tw���o�non-strict�v�ertices�in�v�e�triangles,�rst�conguration�	color pop����	color pop���M��d��src:2347 SevenTriangles.tex�W��:�e�5tak��9e�up�Case�1.�fEac�h�of�the�t�w�o�3-triangle�congurations�exp�A�oses����Wonly���one���side��a��on�its�b�A�oundary��:�.�p�If�the�(or�a)�shared�triangle�is�one����Wof� Ithe�t��9w�o� Itriangles� Hwhose��a��sides�are�on�the�maximal�segmen��9t�in�one����Wconguration,�l9then�A�that�triangle�A�m��9ust�b�A�e�the��T�����3��l��of�the�other�conguration.����WThere��are�just�t��9w�o��suc�h�congurations�p�A�ossible,�|when�����=��Q��R�=�6.�m�When����W��4�=��2Aarctan������P���/�1���/��&�fe�����2�����$�	�,���w��9e�u�will�u�sho�w�no�u�suc�h�conguration�u�is�p�A�ossible:��let��P�H�Q��b�A�e����Wthe���maximal�segmen��9t,�Ɉwith�triangles��T�����1���<�and��T�����2���w��9est�of��P�H�Q��and��T�����3���east�of����W�P�H�Q���with��its��b��side�equal�to��P�H�Q��and�its�righ��9t�angle�at��P�H��.�-Then�the�north����Wside���of����T�����3���~�is�the�only�exp�A�osed�side�of�length��a�,��Wand�it�cannot�o�A�ccur�as�part����Wof�Ta�pair�in�another�conguration,�since�its��b��side�is�already�used.����d��src:2355 SevenTriangles.texTherefore�������=����R�=�6.���W��:�e�will�sho��9w�that�the��t�w�o�congurations�with����W��i��=�h��R�=�6���cannot���b�A�e�completed�to�a�7-tiling.��9Let��P�H�Q��b�A�e�a�(north-south)����Wmaximal��segmen��9t�of�length��	�c�,��and��T�����1�����and���T�����2���w�est��of��P�H�Q��	�with�their��a��sides����Wtogether��matc��9hing���P�H�Q��and�shared�w�est��v�ertex��W�H��;�vthen��T�����3��+/�is�east��of��P�Q�,����Wwith�~�angle�~����T�at��Q��and�east�v��9ertex��E��2�,���and��T�����4���4�shares�side��QE���and�has�angle����W��D��at�B��Q�B��and�east�v��9ertex��X��[�,�N,and��T�����5��mu�shares�side��P�H�X��[�,�N,and�can�b�A�e�placed�in����Weither�"�of�"�t��9w�o�orien�tations.�EcLet��Y�12�b�A�e�the�"�north�v�ertex�of��T�����5��*��.�EcSupp�A�ose,�&dfor����Wpro�A�of�Tb��9y�Tcon�tradiction,�c�that�the�T���D�angle�of��T�����5��~��is�placed�at��X��[�,�c�and�the������W�angle���at��P�H��.�O(See�Figure�11)�Then��W�P�X��[Q����is�a�parallelogram:�>since�angle����W�X��[P�H�Q��E�=��angle��P�QW��,�"��P�X��a�is�parallel��to��W�Q�,�"�and�since�angle��X��[QP�B(�=����Wangle��A�QP�H�W��,��;�QX�Y��is�parallel�to��@�W�P��.��6Since�only�t��9w�o�more��@triangles�can����Wb�A�e�&placed,� Ythe�three�sides�%of�the�nal�triangle�are�among�the�v��9e�sides�of�����>�color push gray 0�����37��Y�	color pop����&@נy�������>�color push gray 0�Y�	color pop���?������W�the��}p�A�en��9tagon��W�H�P�Y�8X��[Q�.���The��}remaining��~t�w�o�triangles�m�ust�therefore�eac�h����Wshare�6�a�side�with�one�of�the�existing�triangles.��LSince��W�H�P���is�parallel�to��X��[Q�,����Ww��9e�Am�ust�Bplace�a�triangle�on�one�of�those�sides.�.8It�is�not�p�A�ossible�to�place����W�T�����6����on��\an��9y�existing�side�in�suc�h�a��]w�a�y�as�to�create�a�conca�v�e�exterior�angle����Wof��y��R�=�3�or�less.��'Hence,��>�T�����6��� �m��9ust�b�A�e�placed�so�as��xto�create�a�quadrilateral�b�y����Wcreating��9straigh��9t��8angles�where�t�w�o�v�ertices��8w�ere�b�A�efore.�There�is��8only�one����Wp�A�osition�o�in�o�whic��9h�that�is�p�A�ossible:��htriangle��T�����6���x�m��9ust�b�e�placed�o�along��P�H�Y����W�with���its�righ��9t�angle�at��Y�8�.��(T��:�ec�hnically�,���w�e���should�compute�the�angle�sums����Wfor��'all�the�other�p�A�ossibly�p�ositions�of��T�����6��*��,��1rather�than�rely�on�insp�ection�of����WFigure��w11�for�a�\pro�A�of��q".)�r�Then��xstraigh��9t�angles�are�created�at��Y����and��P�H��.����WThen���since��W�H�P�j�and��QX�o��are�parallel,��w��9e�m�ust�place��T�����7���-�along��QX��[�.�T��:�o����Wmak��9e�}a�~straigh�t�angle�at�~�X����w�e�need�~to�place�the������angle�of��T�����7��I%�at��X��[�;�#but����Wto��Gmak��9e��Fa�straigh�t��Fangle�at��Q�,��w�e�need��Fto�place�the���Cs�angle�there.��Indeed����Wthis���six-triangle�conguration���can�b�A�e�completed�to�an�8-tiling,��fbut�not�to����Wa���7-tiling.��This���con��9tradiction�sho�ws�that��T�����5���y�cannot���b�A�e�successfully�placed����Wwith�Tits������angle�at��X��[�.��z澍�>�����color push gray 0�mM}���9��src:2379 SevenTriangles.tex����J�A�ٖ̲W����>,����P����>,�D�Q����.I�ٖ�V����S��M]Y����5�ٖ�X����u*���8E��U[��k" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.9226 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.9226 42.67912 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.9226 85.35825 L 0 setlinecap stroke  end �o" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 0.0 moveto 73.9226 85.35825 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 0.0 moveto 147.84457 42.67912 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 0.0 moveto 110.88391 64.01868 L 0 setlinecap stroke  end �v" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 85.35825 moveto 147.84457 42.67912 L 0 setlinecap stroke  end �v" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 85.35825 moveto 110.88391 106.6978 L 0 setlinecap stroke  end �x" tx@Dict begin STP newpath 0.8 SLW 0  setgray  147.84457 42.67912 moveto 110.88391 106.6978 L 0 setlinecap stroke  end �������x��color push gray 0Figure�UU11:�q�Tw���o�non-strict�v�ertices�in�v�e�triangles,�second�conguration�	color pop����	color pop��B��d��src:2385 SevenTriangles.tex�Therefore��the����a<�angle�of��T�����5����is�placed�at��P�H��.���(See�Figure�12).���Then����Wthe�:total�angle�there�is�:3��G)�=����R��,�C>and��W�H�P�Y�HG�is�straigh��9t.���The�:quadrilateral����W�W�H�Y�8X��[Q����is�con��9v�ex.��If���w�e�place�a�triangle�to�share�an�y���of�its�sides,��4and�w�e����Wdo�s"not�s#matc��9h�the�side�exactly��:�,���then�another�triangle�will�b�A�e�required�on����Wthat��side,�Gand��a�large�triangle�cannot�b�A�e�created.�*Hence�when�w��9e�place�a����Wtriangle,�غit�ɒm��9ust�ɓmatc�h�one�ɒof�the�sides�of�the�quadrilateral�exactly��:�.�0That����Wmeans��`that��at��9w�o�of��athe�sides�of�the�quadrilateral�m��9ust�b�A�e�sides�of�the�nal����Wtriangle.��If�
ran��9y�
striangle�is�placed�north�of��W�H�P�Y���then�
sw��9e�m�ust�
rplace�t�w�o����Wtriangles��Jthere,��Msince�the��Klength�of��W�H�P�Y���is��c��%�+��a�.�
That�will��Jlea��9v�e�at��Jleast����Wfour�r�v��9ertices;���hence��W�H�P�Y��-�is�r�one�of�the�sides�of�the�nal�triangle.�5SSince����W�QX���is�e%parallel�to�e$�W�H�P�Y�8�,�yit�e%cannot�b�A�e�a�side�of�the�nal�triangle.��Hence����W�T�����6����m��9ust��Zb�A�e��[placed�south�of��QX��[�,���sharing�side��QX��.�rThere��[are�t��9w�o��Zp�A�ossible����Worien��9tations.�kIf���the���4)�angle�of����T�����6����is�at��X��[�,��then�the�resulting�gure�has����Wv��9ertices�sat��W�H��,�!��Y�8�,��X��[�,�!��Q�,�and�sthe�south�v�ertex�r�Z��[�of��T�����6��*��.�:�It�is�con�v�ex,�!�so�it����Wis��dnot�p�A�ossible�to��ecomplete�it�to�a�triangle�b��9y�placing��T�����7��*��.�� Therefore,���the������W�angle�-oof�-p�T�����6��X�is�at��X��[�.�d�That�creates�a�straigh��9t�angle�is�created�at��Q�,�3vso�the����Wresulting��3gure�is�a�quadrilateral.�No��9w�w�e�ask�if�it��4is�p�A�ossible�to�place��T�����7�����W�along��tan��9y�side�of�this��uquadrilateral�so�as�to�create�a�triangle.�{Sides��W�H�P�Y����W�and��H�W�H�QZ�,/�are�longer�than��G�c�,��~so�it�is�not�p�A�ossible�to�place��T�����7�����there.��lIf��T�����7���is����Wplaced���along����Y�8X��[�,��=then�the���$�angle�w��9ould�go�opp�A�osite��Y�8X��[�,��=the�righ��9t�angle����Ww��9ould��ha�v�e�to��go�at��Y��G�to�a�v�oid�creating��a�fourth�v�ertex�there,�׃lea�ving�the����W�����angle��to��go�at��X��[�,�֧where�the�angle�sum�w��9ould�then�b�A�e�3��sN�+�q���	��=���5����<��R��,�����>�color push gray 0�����38��Y�	color pop����'V�y�������>�color push gray 0�Y�	color pop���?������W�so�J�a�triangle�is�J�not�created.���Hence��T�����7��up�m��9ust�b�A�e�placed�south�of��T�����6��*��.���That����Wmeans��the�sides�of�the�nal�triangle�are��W�H�Y�8�,�b
�Y�X����extended,�and���W�H�Q����W�extended.�y�Let���Z����b�A�e��the�in��9tersection�p�oin��9t�of�lines��W�H�Q���and��Y�8X��[�.�y�This����Wp�A�oin��9t��lies�to�the��southeast,��b�ecause�of�the�angles��made�b��9y�the�transv�ersal����W�W�H�Y�8�.�Z)T��:�riangle�Ԓ�Z���X��[Q�ԑ�is�similar�to�the�tile,�ab�A�ecause�it�has�angle����X�at��Z����W�(b�A�ecause�{Uthe�angle�sum�{Tof�triangle��W�H�Y�8Z�=�m��9ust�b�e���R��),���and�{Tangle�����at��Q����W�(b�A�ecause�~��W�H�QZ�"��is�a�straigh��9t�angle).��KBut�it�has�side�~��QX�F!�=����c��=�2�a�~��opp�osite����Wangle�Y�����.��Hence�Y�its�area�is�four�times�that�of�the�tile,�j�and�th��9us�it�cannot����Wb�A�e�
�tiled�
�b��9y��T�����6��5��and��T�����7��*��.��That�con�tradiction�
�completes�the�pro�A�of�of�Case�1.����d��src:2410 SevenTriangles.texNo��9w�LRw�e�LStak�e�up�LSCase�2,�Zin�whic�h�LSthe�t�w�o�LScongurations�share�a�side����Wbut�%Jnot�%Ia�triangle.�LQWhen�w��9e�join�t��9w�o�con�v�ex�%Jquadrilaterals�along�a�side,����Ww��9e��>get��=a�gure�with�at�least�six�con��9v�ex��>v�ertices,���and�p�A�ossibly��=with�one����Wor���t��9w�o���more�v�ertices,��
whic�h�migh�t���b�A�e�conca�v�e.�r�Placing�one���triangle�can����Wreduce�%#the�%"n��9um�b�A�er�of�con�v�ex�%"v�ertices�b�y�at�%"most�2�(that�can�happ�A�en�if�a����Wconca��9v�e�A`v�ertex�A_o�A�ccurs�in�just�the�righ��9t�p�A�osition).���But�ev�en�if�A_that�happ�A�ens,����Wthere�{�will�{�still�b�A�e�four�v��9ertices�left�after�placing��T�����7��*��.�PBThat�completes�the����Wpro�A�of�Tin�Case�2.����d��src:2416 SevenTriangles.texFinally��w��9e��consider�Case�3,�N&in�whic��9h�the�t�w�o�congurations��do�not����Wev��9en��share�a�side.��Since�the�quadrilaterals�ѿare�con�v�ex,��Dthey�share�at�most����Wone���p�A�oin��9t.�f�A�t�that�shared�p�A�oin�t�there�ma�y�b�A�e�a���v�ertex�with�a�conca�v�e����Wexterior���angle,���so�placing���one�triangle�could�p�A�ossibly�eliminate�t��9w�o�of���the����Wsix�A�remaining�A�v��9ertices,��but�that�w��9ould�still�lea��9v�e�four,��to�A�o�man�y�A�for�a����Wtriangle.�pThat�Tcompletes�the�pro�A�of�of�the�lemma.������R�color push gray 0��Theorem���5�(Main�Theorem)�	color pop�����_�src:2421 SevenTriangles.tex�Ther��e�N<is�no�7-tiling.����W�src:2424 SevenTriangles.texPr��o�of�.��}Supp�A�ose�Atriangle�A�AB�r�C����is�7-tiled�b��9y�sev�en�Acopies�of�triangle��T�H��.����WThen�0�according�to�0�our�previous�theorems,�w�it�is�not�a�strict�tiling,�w�and����Wthere��@is�exactly��?one�non-strict�v��9ertex��V�8�,��wand�triangle��T�5#�has�a�righ��9t�angle,����Wand�l�its�small�l�angle���n��is�either���R�=�6�or��arctan��#�(1�=�2),�¾and�the�non-strict����Wv��9ertex�8o�A�ccurs�in�8one�of�t�w�o�sp�A�ecic�congurations�of�8three�triangles�(one����Wfor�Okeac��9h�����).�ʴ(Those�congurations�are�illustrated�in�Figure�8.)�T��:�o�nish����Wthe��ypro�A�of,��w��9e�ha�v�e��xto�sho�w�that��xit�is�imp�A�ossible,��starting�from�either�of����Wthose��'congurations,�
�to�add�four�more�copies�of��T�%
�to�create�a�triangle.����WW��:�e�*�need�only�consider�placemen��9ts�of�*�new�copies�of��T�s��that�share�sides����Wwith��Sexisting�copies,��since�no�additional�non-strict�v��9ertices��Tcan�o�A�ccur�in�a����W7-tiling.����d��src:2432 SevenTriangles.texThere�{�are�{�7���Z�angles�total�in�the�sev��9en�copies�of��T�H��.��9Of�these,��k���Y�are�used����Wat�tthe�v��9ertices��A�,���B�r��,�and�t�C����,�and���S�at�the�non-strict�v��9ertex��V�8�,�lea��9ving�5�����W�to�(mb�A�e�(lused�at�b�oundary�(land�in��9terior�v�ertices�(l(other�than��V�8�).�U�An�in��9terior����Wv��9ertex��uses��2��R��,�Wand�a�b�A�oundary�v��9ertex�uses���R��.�!BThe�p�A�ossibilities�are�th��9us:����Wone�@�b�A�oundary�and�@�t��9w�o�@�in�terior�v�ertices,�k-or�three�b�A�oundary�@�and�one�in�terior����Wv��9ertex,�_sor�1�v�e�1�b�A�oundary�v�ertices�and�no�in�terior�v�ertex.�ШIn�particular�there����Ware�Tat�most�t��9w�o�Tin�terior�v�ertices.����d��src:2441 SevenTriangles.texFirst�[�w��9e�[�tak�e�up�[�the�case���	?�=�w��R�=�6.��IThe�starting�conguration�is�the����Wrst��done��esho��9wn�in�Figure�8.�vThe�non-strict�v��9ertex��V�۝�is�at�the�midp�A�oin��9t�of����Wnorth-south�!line��P�H�Q�.�?�T��:�riangles��T�����1��K��and��T�����2���are�w��9est�of�!�P�H�Q�,�$with�a�shared����Ww��9est��@v�ertex��?�W�H��,��a�righ��9t�angle�at�their�shared�v��9ertex��V�8�,��and�angle�����at��W�H��.����WT��:�riangle�T�T�����3��?��is�east�of��P�H�Q�,�with�angle��P�QE���=�������,�and�angle��QP�E���=�����x,�.����d��src:2447 SevenTriangles.texConsider���adding��T�����4��ɤ�north�of��P�H�E�".�with���its�third�v��9ertex��N����on��QE��ex-����Wtended.��}If�^zw��9e�^{then�add��T�����5���!�north�of��P�H�N����,��
t��9w�o�^zadditional�triangles��T�����6���"�and��T�����7������>�color push gray 0�����39��Y�	color pop����(mY�y�������>�color push gray 0�Y�	color pop���?������W�will��b�A�e�required��to�ll�the�angle�2��mZ�at��P�H��.�,�V��:�ertices��Q��and��N���remain,�so�if����Wthis��'w��9ere�to�create��&a�triangle,��0the�w�est�v�ertex��Y��_�w�ould�ha�v�e��&to�lie�on��QW����W�extended.��That�yw��9ould�require�xat�least�t�w�o�more�triangles�to�share�v�ertex����W�W�H��,��Wone��Wof�whic��9h�migh�t�b�A�e��T�����6��*��,��Wbut�there�is��Xno�second�one�a�v��|railable.��qHence����Wthe��Zindicated�placemen��9t�of��Y�T�����5����fails.�u�Similarly��:�,��if�w�e��Yadd��T�����5����southeast�of����W�E��2N����,��Ywith��southeast�v��9ertex���X��[�,�then�the�exterior�angle�at��v��9ertex��E�`L�will�b�A�e����Wconca��9v�e,�so��w�e�will��ha�v�e�to�add��T�����6��9W�sharing��v�ertex��E��2�.�:That��can�ll�v�ertex����W�E����to�'Y2��z�only�if�b�A�oth��T�����5��R�and��T�����6��Q��ha��9v�e�'Ya�righ��9t�angle�at�'X�E��2�,�+�whic�h�will�mak�e����Wa��six-triangle��con��9v�ex�p�A�en�tagon;��suc�h�a�conguration�cannot��b�A�e�completed����Wto�%.a�%-triangle.�K�If�instead��T�����5��*��,�)$�T�����6���,�and�%.�T�����7��O��are�all�%-placed�with�a�v��9ertex�at��E��2�,����Wthe�e�result�cannot�e�b�A�e�a�triangle�since�there�are�v��9ertices�at��W�H��,�y��N����,��Q�,�and����Wat�|�least�one�more�southeast�of��QN����.�R^Hence�it�fails�to�place��T�����5���J�southeast����Wof�#3�E��2N����.�FSince��T�����5��M��cannot�#2b�A�e�placed�east�of��E��2N��or�north�of��P�H�N����,�&�with�this����Wplacemen��9t�DFof�DE�T�����4��*��,�Pt�w�o�sides�of��AB�r�C��-�m�ust�b�A�e��W�H�P�N�=)�and��N���E��2Q�.��EW��:�e�m��9ust����Wthen�<add�the�;remaining�three�triangles�to�the�south��9w�est�<of��W�H�Q�.�Since��P����W�and�z>�E��q�will�no��9w�z?b�A�e�b�oundary�z?v��9ertices,��Cw�e�are�z>allo�w�ed�only�z>one�in�terior�z>v�er-����Wtex���in�the���pro�A�cess.���That�v��9ertex��S�0��will�b�A�e�created�when�w��9e�add�triangle��T�����5�����W�immediately��south�of���W�H�Q�,�7with�south��9w�est�v�ertex���S���.��There�are�t��9w�o��w�a�ys����Wto���place��T�����5��*��;��1rst�consider�placing�its������angle�at��Q�.��)Then��W�H�S�k�=��f�a��and����Ww��9e���m�ust�place��T�����6���=�w�est�of��W�H�S���with�its�righ�t�angle���at��S���(since�otherwise����W�W�H�S���m��9ust�H�b�A�e�H�the�third�side�of�the�nal�triangle,�U�and�there�is�not�enough����Warea�Wasouth�Wbof��S��Q��and�north�of��N���Q��extended�to�hold�t��9w�o�Wamore�copies�of����Wthe�X7tile).��Let�X6�X���b�A�e�the�w��9est�v�ertex�X6of��T�����6�����(placed�w��9est�of��W�H�S���).��W��:�e�no��9w����Wha��9v�e��ha�non-strict�6-tiling�of�triangle��X��[N���Q��i�(Figure�13).���This�cannot�b�A�e����Wmade�_9in��9to�a�triangle�b�y�adding�one�_:more�triangle�south�of��X��[S��Q�.��Hence����Wthe�Tindicated�placemen��9t�of��T�����5��?��fails.����X��>��h��color push gray 0���X���9��src:2471 SevenTriangles.tex����u^��R%�W�������Y@P����0���#Q�����%�ٖ�V���37��#X����	���#S���j���oN����EY���E���	��k" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.9226 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.9226 42.67912 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.9226 85.35825 L 0 setlinecap stroke  end �o" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 0.0 moveto 73.9226 85.35825 L 0 setlinecap stroke  end �k" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 0.0 moveto -73.9226 0.0 L 0 setlinecap stroke  end �s" tx@Dict begin STP newpath 0.8 SLW 0  setgray  -73.9226 0.0 moveto 147.84457 128.03737 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 0.0 moveto 147.84457 128.03737 L 0 setlinecap stroke  end �g" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 0.0 moveto 0.0 42.67912 L 0 setlinecap stroke  end �v" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 85.35825 moveto 110.88391 64.01868 L 0 setlinecap stroke  end �������&8��color push gray 0Figure�UU12:�q�A�six-tiling�that�cannot�b�Ge�completed�to�a�7-tiling�	color pop����	color pop��@��d��src:2477 SevenTriangles.tex�Therefore�mx�T�����5����has�to�b�A�e�mwplaced�with�its�����angle�at��Q�,���and�its���o?�angle����Wat�+�W�H��.�Let��S��0�b�A�e�,the�third�v��9ertex�of��T�����5��*��.�Then�w�e�cannot�,place��T�����6��5��with�its����W���|�angle�۵at��Q��to�ha��9v�e�۵its�side�extend��N���E��2Q�,��;since�that�creates�a�non-strict����Wv��9ertex�g�at��S���.�Hence��QS���m�ust�b�A�e�g�the�third�side�of�the�nal�triangle.�But����Wthat�[6is�[5not�p�A�ossible,��oeither,�since�[6�QS��:�is�parallel�to��W�H�P�N�T�(since�[6transv��9ersal����W�P�H�Q�*��mak��9es�equal�*�alternate�in�terior�angles��S��QP�s��and��QP�H�N����,�0Xb�A�oth�equal�to����W2��x,�).�'this��placemen��9t��of��T�����5��C��also�fails.�Hence�the��indicated�placemen��9t�of��T�����4�����W�(north�Tof��P�H�E����with�its�third�v��9ertex��N�8�on��QE��extended)�fails.�����>�color push gray 0�����40��Y�	color pop����)���y�������>�color push gray 0�Y�	color pop���?������d��src:2485 SevenTriangles.tex�No��9w���consider�adding��T�����4���|�north�of��P�H�E�@�with�is�third�v�ertex��N����not�on����W�E��2Q�*/�extended,�/ei.e.�[its�*.righ��9t�angle�is�at��P�s�instead�of��E��2�.�[Then�the�exterior����Wangle�
fat�v��9ertex�
g�P�VI�is�conca�v�e,��with�
gtotal�in�terior�angle�7��R�=�6.��A�t�least�t�w�o����Wmore��%triangles��T�����5�����and��T�����6���m��9ust�share�v�ertex��P�H��.��If�w�e�use��T�����7�����also�at��P�H��,����Wthen���w��9e�will���still�ha�v�e�v�ertices�four�v�ertices����W�H��,��Q�,��E��2�,�and����N����,�so�w��9e�m�ust����Wuse��only��T�����5��&��and��T�����6���at��P�H��.��W��:�e�m��9ust�therefore�place��T�����5���along��N���P�ک�=����b��with����Wits�Yrigh��9t�angle�at�Y�P�H��.��Let��Y�gL�b�A�e�its�third�v��9ertex.��Then��T�����6�����m�ust�Yb�A�e�placed����Wwith���its��c��side�along��W�H�P����and�its���+�angle�at����P��,���so��Y�8P��q�=�z��a��is�matc��9hed.����WThe�Tangle�of��T�����6��~��at��Y�b?�is���R�=�2,�c�so�the�total�angle�Tat��Y��is�5��R�=�6,�c�not����,�c�and����Wour�:I6-triangle�conguration�has�v��9ertices�:Hat��W�H��,�C��Y�8�,��N����,��E��2�,�and�:I�Q�.��O(Figure����W14.)�	�Since�d}this�d~conguration�is�con��9v�ex,�xHthe�d~b�A�est�w��9e�could�hop�A�e�to�do�b��9y����Wplacing���T�����7��G��is�to�reduce�the��n��9um�b�A�er��of�v��9ertices�b�y��one�to�four.�3ZHence�this����Wplacemen��9t�Tof��T�����4��?��also�fails.����X��>��h��color push gray 0���X���9��src:2497 SevenTriangles.tex����X�5��R%�W�������Y@P�����Ɵ�#Q����h=�ٖ�V�����q��j��N�����+��M]Y�����q���E��c���k" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.9226 0.0 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.9226 42.67912 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 73.9226 85.35825 L 0 setlinecap stroke  end �o" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 0.0 moveto 73.9226 85.35825 L 0 setlinecap stroke  end �p" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 42.67912 moveto 36.9613 106.6978 L 0 setlinecap stroke  end �w" tx@Dict begin STP newpath 0.8 SLW 0  setgray  36.9613 106.6978 moveto 110.88391 149.37692 L 0 setlinecap stroke  end �w" tx@Dict begin STP newpath 0.8 SLW 0  setgray  73.9226 85.35825 moveto 110.88391 149.37692 L 0 setlinecap stroke  end �t" tx@Dict begin STP newpath 0.8 SLW 0  setgray  36.9613 106.6978 moveto 73.9226 85.35825 L 0 setlinecap stroke  end �y" tx@Dict begin STP newpath 0.8 SLW 0  setgray  110.88391 64.01868 moveto 110.88391 149.37692 L 0 setlinecap stroke  end �v" tx@Dict begin STP newpath 0.8 SLW 0  setgray  110.88391 64.01868 moveto 73.9226 85.35825 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  110.88391 64.01868 moveto 73.9226 0.0 L 0 setlinecap stroke  end �������x��color push gray 0Figure�UU13:�q�Another�conguration�that�cannot�b�Ge�completed�to�a�7-tiling�	color pop����	color pop��@��d��src:2502 SevenTriangles.tex�Th��9us����T�����4���L�cannot�b�A�e���placed�north�of��P�H�E�4��in�either�orien��9tation.��eHence����W�P�H�E���lies���on���the�b�A�oundary�of�triangle��AB�r�C����.��6Since��W�Q��is�parallel���to��P�E��2�,���it����Wfollo��9ws�/#that��W�H�Q��is��not��a�side�of�/$triangle��AB�r�C����.�i�W��:�e�m�ust�therefore�place����W�T�����4���T�south�ͬof�ͭ�W�H�Q�;�)�call�its�third�v��9ertex��S���.�EzThere�are�t��9w�o�orien�tations�ͭto����Wconsider:��Keither��Athe��Bangle�of��T�����4�� ��at��Q��is���nm�or�it�is�����.��8First�supp�A�ose��T�����4�����W�has���angle������at��Q�.��
Then��S��W���is�a�north-south�line.�Consider�placing��T�����5�����W�along����W�H�P����with������at��P��.��Let��X�Z<�b�A�e�the���third�v��9ertex�of��T�����5��*��.��Then��X��[P����is����Wparallel��to��S��Q�,��so��T�����6��:&�m��9ust�b�A�e�placed�either�north�of��X��[P�Xb�or�south�of��S��Q�.����WIn��oeither�case��nw��9e�are�committed�to�making��W�H�S��t�extended�a�side�of��AB�r�C����,����Wsince�¶placing�another�triangle�w��9est�of��W�H�S�E��extended�will�create�conca�v�e����Wv��9ertices.�u�W��:�e��pm�ust�therefore�denitely�add��o�T�����6����north�of��X��[P��S�to�reac�h�the����Win��9tersection�I�p�A�oin�t�I��Y�X)�of��P�H�E��#�and��W�S���extended.��HNo��9w�I�w�e�ha�v�e�I�v�ertices��Y�8�,����W�E��2�,���Q�,�and����S���,�and�w��9e���m�ust�remo�v�e���the�v�ertex�at����Q��b�y�placing����T�����7��*��.��LThe����Wangle�B�at��Q��is�B�presen��9tly�2����+��@����=��[2��R�=�3.��T��:�o�remo�v�e�B�it�w�e�w�ould�ha�v�e����Wto��Vput�the��U��7��angle�of��T�����7����at��Q�,���but�the�t��9w�o��Vsides��S��Q��and��QE�B��are�b�A�oth����W�b�,��Mso���the�����$��angle�of��T�����7���)�cannot�b�A�e�placed�at��Q�.��Hence�the�placemen��9t�of����W�T�����5��C��along���W�H�P�a��with������at��P��fails.�@1No��9w��consider�placing��T�����5��C��along��W�H�P����W�with�����/�at��P�H��.�9Then�the�third���v��9ertex��X�kG�of��T�����5����lies�on��P�H�E�:�extended,��Rand����W�X��[W����is�tp�A�erp�endicular�to��P�H�E��2�.�8�If�w��9e�place��T�����6�����along��X��[W��,���w��9e�again�reac�h����Wthe��3in��9tersection��2p�A�oin�t��Y��j�of��P�H�E�)d�and��W�S���,��iand�w��9e��2ha�v�e�the��2same�con�v�ex����Wquadrilateral�xx6-tiled�xyas�with�the�previous�placemen��9t�of��T�����5��*��,��Aand�again��T�����7������>�color push gray 0�����41��Y�	color pop����*���y�������>�color push gray 0�Y�	color pop���?������W�cannot�Nb�A�e�Mplaced�to�mak��9e�a�triangle.��]Hence�w��9e�cannot�place��T�����6��2��along����W�X��[W�H��.��But�zthen�y�X�W�]\�m��9ust�b�A�e�ya�side�of�the�nal�triangle��AB�r�C����.��Since����W�P�H�E���is�?�a�?�side,�Jsthe�v��9ertex�at��W����will�ha��9v�e�to�?�b�A�e�eliminated,�Jsbut�this�is�not����Wp�A�ossible,�*5since�&w��9e�w�ould�&ha�v�e�to�place�angle���'��at��W�H��,�*5but�&the�side��W�S��
�is����W�a�,��whic��9h���cannot���b�A�e�adjacen�t���to�angle�����.�
NHence�the�placemen��9t�of��T�����5����along����W�W�H�P�	@�with��]��8��at��^�P��fails.��No��9w�b�A�oth�p�ossible�placemen��9ts��^of��T�����5����along��W�H�P����W�ha��9v�e�'�failed.�SMHence��W�H�P�p��is�con��9tained�'�in�one�of�the�sides�of�triangle��AB�r�C����.����WW��:�e�s�therefore�s�m��9ust�add��T�����5���b�w��9est�of��W�H�S���,��Tmatc��9hing�its��a��side�to��W�H�S���.�7�Let����W�X�|��b�A�e�Ȇthe�third�ȅ(w��9esternmost)�v�ertex�of��T�����5��*��.��If��X��[S��Q��is�not�ȅa�side�of��AB�r�C����,����Ww��9e�1Xw�ould�1Yha�v�e�to�1Yadd�t�w�o�1Ymore�triangles�south�of��X��[S��Q�,�8Ybut�that�w��9ould����Wnot��mak��9e�a�triangle.���Hence��X��[S��Q��is��the�third�side�of��AB�r�C����.�Let��Y��Q�b�A�e����Wthe�^�in��9tersection�^�p�A�oin�t�of�^��E��2P����and��S��Q�.���Then�the�remaining�t��9w�o�triangles����Ww��9ould�K�ha�v�e�to�tile�triangle�K��E��2QY�8�.��/T��:�riangle��E�QY�Z!�is�similar�to�K��T�H��,�Y�since�it����Whas��angle���:�at��Q���and�a�righ��9t�angle�at��E��2�,�Ǽbut�the�side�opp�A�osite�angle������W�is��"�E��2Q�a��=��b��=������?�p������?�aH���u���3������=�2.���So�the�area�of�triangle��E��2QY��Z�is�3�times�the�area�of����W�T�H��,�#not��t��9wice��the�area�of��T�H��.��Hence�the�placemen��9t�of��T�����4��/~�with�angle�����at��Q����W�fails.����d��src:2532 SevenTriangles.texNo��9w��
consider��the�other�p�A�ossible�placemen��9t�of��T�����4��*��,���namely�south�of��W�H�Q����W�with���angle�������at��Q�.���Let��S�"��b�A�e�the�third�v��9ertex�of��T�����4��*��.���If��W�H�S�"��is�a�side�of����W�AB�r�C����,��then��Lw��9e��Kwill�need�to�use�three�more�triangles�north�of��W�H�P��/�to�reac��9h����Wthe���in��9tersection�p�A�oin�t��Y�
�of��W�H�S�~��and��E��2P��,��and�that���will�lea��9v�e���four�v��9ertices����W�Y�8�,���S���,���E��2�,�and�u#�Q�.�;�Hence�u$�W�H�S��(�is�not�a�side�of��AB�r�C����,��so�w��9e�m�ust�place��T�����5�����W�along�7C�W�H�S�MY�=��T�b�.��=There�7Bare�t��9w�o�7Cp�A�ossible�orien��9tations,�?�with�the�angle�of��T�����5�����W�at�r�W����either���s��or�a�righ��9t�angle.�2�Consider�rst�placing��T�����5�����on��W�H�S���with�a����Wrigh��9t�L�angle�L�at��W�H��.��^Let��X�T�b�A�e�the�third�v��9ertex�of��T�����5��*��.��]Then��X�S�lies�on��W�H�P����W�extended�Sand��X��[S��X�is�parallel�to��W�H�Q��and��P�E��2�.�Then��X��[S��X�cannot�b�A�e�a�side����Wof���AB�r�C����(since��P�H�E�_��is��a�side),��so�w��9e�m�ust��place��T�����6��g�south�of��X��[S���.�	�Let��Y���b�A�e����Wthe��third��v��9ertex�of��T�����6��*��.��Then�w��9e�ha�v�e��v�ertices��Y�8�,����X��[�,��P�H��,��E��2�,��Q�,�and��p�A�ossibly����W�S���.�h�Since��this��gure�is�con��9v�ex,���w�e��cannot�p�A�ossibly�reduce�the�n��9um�b�A�er�of����Wv��9ertices��sto��rthree�b�y�placing��r�T�����7��*��.�V�So�the�placemen��9t�of��T�����5����on��W�H�S�Vw�with�a����Wrigh��9t�C=angle�C<at��W�� �fails.��*No�w�C<consider�the�other�p�A�ossible�placemen��9t�of��T�����5��*��,����Won�kM�W�H�S��Q�with�angle�kL��m�at��W��.�YLet��X���b�A�e�kLthe�third�v��9ertex�of��T�����5��*��.�YNo�w��W�H�X����W�cannot���b�A�e�a���side�of��AB�r�C����,���since�in�that�case�three�more�triangles�w��9ould�b�A�e����Wneeded�h
to�h	reac��9h�the�in��9tersection�p�A�oin�t�h	�Y�vB�of��P�H�E��<�and��W�X��[�.��Therefore�w��9e����Wm��9ust��add���T�����6��I��w�est�of���W�H�X��[�.�9iLet��Z����b�A�e�the�w��9esternmost�v�ertex��of��T�����6��*��.�9iThen����Ww��9e���ha�v�e�v�ertices��Z����,��r�X��[�,��s�Q�,��E��2�,��P���at���least,�and���the�gure�is�con��9v�ex,��rso���it����Wcannot��-b�A�e��.made�in��9to�a�triangle�b��9y�placing��T�����7��*��.��Hence�b�A�oth�orien��9tations�of����W�T�����5����on����W�H�S�D��fail.��Hence�the�second�p�A�ossible�orien��9tation�of����T�����4���(south�of��W�H�Q����W�with��"angle��#��N�at��Q�)�fails.�t�That�exhausts�the�p�A�ossibilities,���and�completes����Wthe�Tpro�A�of�in�case������=�����R�=�6.����d��src:2550 SevenTriangles.texNo��9w��7w�e�consider�the�case������=����arctan������P����1�����&�fe�����2�����#��,��whic�h�is�ab�A�out�26.565�degrees.����WThen�/���6`�=���4arctan����2�/�is�ab�A�out�63.435�degrees.�liThe�starting�conguration�is����Wsho��9wn�u�in�the�u�second�part�of�Figure�8.�=The�non-strict�v�ertex��V����is�at�the����Wmidp�A�oin��9t��2of�north-south�line��P�H�Q�.��T��:�riangles��T�����1�����and��T�����2���are�w��9est�of��P�H�Q�,����Wwith�la�mshared�w��9est�v�ertex�l�W�H��,�	ha�righ��9t�angle�at�their�shared�v��9ertex��V�8�,�	hand����Wangle�Ä���K�at��W�H��.�+T��:�riangle��T�����3���+�is�east�of��P�Q�,���with�angle��P�QE���=�������,���and�angle����W�QP�H�E����is�Ta�righ��9t�angle.����d��src:2556 SevenTriangles.texW��:�e��rst�consider��placing��T�����4��K�north�of��P�H�E�v��with�a�righ��9t�angle�at��P�H��.�5Let����W�N�KM�b�A�e�Rhits�Rinorthern�v��9ertex.��wThat�creates�a�conca��9v�e�v�ertex�Rhat��P��K�with�exterior����Wangle������'��v}��x,�.��That�will�require��T�����5���b�and��T�����6���to���b�A�e�used�north�of��W�H�P����and�����>�color push gray 0�����42��Y�	color pop����+���y�������>�color push gray 0�Y�	color pop���?������W�w��9est��pof��o�N���P�H��,resp�A�ectiv�ely��:�.�$Ev�en�if��ow�e�managed��oto�solv�e��othe�problem�of�the����Wconca��9v�e�ޠv�ertex�ޟat��P�H��,��w�e�ޟw�ould�then�ha�v�e�ޟonly�one�more�triangle��T�����7��	G�to����Wplace,��and��Hw��9e��Icannot�reduce�the�n�um�b�A�er��Hof�v�ertices�b�y�placing��Hit�on��N���E��2�,����W�QE��2�,�<�or���W�H�Q�,�but��it�m��9ust�b�A�e�placed�on�one�of�those�sides�since�not�all����Wthree�E�can�E�b�A�e�sides�of�the�nal�triangle��AB�r�C����.��yHence�the�placemen��9t�of��T�����4�����W�north�Ӡof�ӡ�P�H�E�V��with�a�righ��9t�angle�at��P���fails.��Next�consider�placing��T�����4���G�north����Wof��O�P�H�E�?��with�a�righ��9t�angle��Nat��E��2�,��
and�let��N��3�b�A�e�its�northern�v��9ertex.�`This����Walso��5creates�a�conca��9v�e��5v�ertex�at��4�P�H��,�-with�exterior�angle�3��R�=�2��y���x�2��x,�.�tW��:�e����Wcannot��use��all�three�remaining�triangles�at��P�H��,���as�that�will�lea��9v�e��v�ertices����W�W�H��,��n�Q�,��o�E��2�,�and���N����.��Since���w��9e�can�place�only�t��9w�o��new�triangles�with�v��9ertices����Wat��,�P�H��,��they�m��9ust�ha�v�e�their��c��sides��+along��W�H�P��and��P�N����,��so�they�cannot����Wha��9v�e�E+righ�t�angles�at��P�H��.���Therefore�their�maxim�um�con�tribution�to�the����Wangle��[sum��\at��P�>�is�2��x,�,��\whic��9h�is�not�enough�to�ll�the�angle�at��P�H��,��]since����W4����+�����R�=�2��Qis�ab�A�out�343.74�degrees,�Pnot�360.�tgHence�the�placemen��9t�of��T�����4�����W�north��of��P�H�E�h2�with�a�righ��9t�angle�at��E��fails.��wHence��T�����4����cannot�b�A�e�placed����Wnorth��.of��-�P�H�E�H`�at�all.��Hence��P�E�H`�is�(con��9tained��.in)�a�side�of�the�nal�triangle����W�AB�r�C����.�
$Supp�A�ose,��for��rpro�of��qb��9y�con�tradiction,��that��W�H�Q��is��ra�side�of�the�nal����Wtriangle.���Then��"w��9e��!m�ust�place��T�����4�����north�of��"�W�H�P��,�Ɣand��T�����5�����w��9est�of��"�T�����4��*��.���Let����W�X���b�A�e���the��w��9est�v�ertex�of��T�����5��*��.���Then��X��[QE�s��is��5-tiled.���W��:�e�m�ust���place��T�����6�����W�along��~�QE��2�,��Isince��X��[W�H�Q��and��X�P�H�E�7��are�sides�of�the��nal�triangle.���There����Ware��t��9w�o��p�A�ossible�orien�tations��of��T�����6��*��,�۾with�angle������at��E�7B�or�angle���,;�at��E��2�.����WFirst�q�consider�q�placing��T�����6���3�on��QE���with�angle���sS�at��E��2�.�1Let��Y���b�A�e�the�third����Wv��9ertex��of��T�����6��*��.��Since��QE���=����c�,���the��angle�of��T�����6����at��Y�W�is�a�righ��9t�angle�and��QY����W�is���parallel���to�side��P�H�E�(��of�the�nal�triangle.��5Hence��T�����7���M�m��9ust�b�A�e�placed�south����Wof�f��QY�(�=���a�,�{Wand�its�righ��9t�f�angle�m�ust�go�at��Y�u(�or�a�v�ertex�will�b�A�e�created����Wthere.��Hence��8the��7angle�of��T�����7�����at��Q��will�b�A�e���x,�,��pmaking�the�total�angle�at����W�Q��_�equal��^to�3���k�+�x?�����=�����R�=�2�+�2��>��R��,��!con��9tradicting��_our��^assumption�that����W�W�H�Q����is�a�side�of�the���nal�triangle.��Hence�the�placemen��9t�of��T�����6����on��QE�F"�with����Wangle�[i��]1�at��E�ޜ�fails.��Next�consider�the�[jother�p�A�ossible�placemen��9t�of��T�����6��*��,�l�on����W�QE�@��with��fangle��e��5��at��E��2�.��Let��Y�˝�b�A�e�the�eastern�v��9ertex�of��T�����6��*��.��W��:�e�cannot����Wplace����T�����7�����north�of����E��2Y�8�,���since�3���Y>�J.�֢�and��T�����7�����w��9ould�then���extend�north�of����W�P�H�E��2�.�(�Hence�n�the�n�third�side�of��AB�r�C���m��9ust�b�A�e��E��2Y�8�,��0and��T�����7���x�m��9ust�b�e�n�placed����Wsouth���of��QY���with�its�righ��9t�angle�at��Y�8�.��Then�the�angle�of��T�����7��։�at��Q��is������W�and��|the��}total�angle�at��Q��is�3���o�+�����d��=���R�=�2�+���2���&<��^��,�"Gso��|a��}triangle�has����Wnot��-b�A�een�created.�k�Hence�the�placemen��9t�of��T�����6�����on��QE�_�with�angle����Y�at��E����W�fails.��5No��9w���b�A�oth�p�ossible�placemen��9ts�of����T�����6���=�ha�v�e�failed.��5This�con�tradicts����Wour��Lassumption��Kthat��W�H�Q��is�a�side�of�the�nal�triangle.��Hence��W�H�Q��is�not����Wa�Tside�of�the�nal�triangle.����d��src:2599 SevenTriangles.texTherefore���w��9e�m�ust�place����T�����4���v�south�of��W�H�Q�.���First�consider�placing��T�����4�����W�along���W�H�Q���with�angle����q�at��Q�,���as�sho��9wn�in�the�rst�part�of�Figure�15.��Let����W�R����b�A�e�qfthe�qgthird�v��9ertex�of��T�����4��*��.���Supp�A�ose,��0for�pro�of�b��9y�qfcon�tradiction,��0that��R�>W����W�is�-�a�-�side�of�the�nal�triangle.�e�Then�w��9e�m�ust�-�place��T�����5��Xz�north�of��W�H�P��.�e�Call����Wits�eXnorth�v��9ertex��N����.�{Since��R�>Q��is�parallel�to��P�H�E��2�,�yY�R�Q��is�eWnot�a�side�of�the����Wnal���triangle,��and���w��9e�m�ust�place����T�����6���V�south�of��R�>Q�.���Call�its�south�v��9ertex����W�X��[�.��Since���P�H�E����is�(con��9tained��in)�a�side�of�the�nal�triangle,�
pthe�east�v�ertex����Wof�xthe�nal�triangle�ym��9ust�lie�on��P�H�E����(extended);��but�since�the�exterior����Wangle��?b�A�et��9w�een��E��2Q��>�and��P�H�E�q�extended�is�more�than���R�=�2,���triangle��T�����7�����cannot����Wextend��to�the�east�of��E�m��on��P�H�E��extended.�/Hence��E��is�a�v��9ertex�of�the�nal����Wtriangle.�	It��is��not�p�A�ossible�to�create�a�nal�triangle�b��9y�placing�triangle��T�����7�����W�along��*the��)east�side�of��E��2Q�,��_since�then�w��9e�will�ha��9v�e��*four�distinct�v��9ertices�����>�color push gray 0�����43��Y�	color pop����,�^�y�������>�color push gray 0�Y�	color pop���?���q�Y��>��h��color push gray 0���Y���9��src:2591 SevenTriangles.tex����ꙟ��R���[���7W���m����y.P����}y���E���t_$��Q���t_$���7V��s��y" tx@Dict begin STP newpath 0.8 SLW 0  setgray  85.35826 110.96599 moveto 128.03738 110.96599 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 68.28687 moveto 85.35826 68.28687 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 68.28687 moveto 85.35826 110.96599 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 68.28687 moveto 85.35826 25.60773 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 25.60773 moveto 85.35826 25.60773 L 0 setlinecap stroke  end �w" tx@Dict begin STP newpath 0.8 SLW 0  setgray  85.35826 25.60773 moveto 85.35826 110.96599 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 68.28687 moveto 0.0 25.60773 L 0 setlinecap stroke  end �x" tx@Dict begin STP newpath 0.8 SLW 0  setgray  85.35826 25.60773 moveto 128.03738 110.96599 L 0 setlinecap stroke  end ���������+�R����،���7W���
o��y.P���8���E���ܝ��Q���ܝ���7V����`�y" tx@Dict begin STP newpath 0.8 SLW 0  setgray  85.35826 110.96599 moveto 128.03738 110.96599 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 68.28687 moveto 85.35826 68.28687 L 0 setlinecap stroke  end �r" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 68.28687 moveto 85.35826 110.96599 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 68.28687 moveto 85.35826 25.60773 L 0 setlinecap stroke  end �q" tx@Dict begin STP newpath 0.8 SLW 0  setgray  51.21547 0.0 moveto 85.35826 25.60773 L 0 setlinecap stroke  end �w" tx@Dict begin STP newpath 0.8 SLW 0  setgray  85.35826 25.60773 moveto 85.35826 110.96599 L 0 setlinecap stroke  end �l" tx@Dict begin STP newpath 0.8 SLW 0  setgray  0.0 68.28687 moveto 51.21547 0.0 L 0 setlinecap stroke  end �x" tx@Dict begin STP newpath 0.8 SLW 0  setgray  85.35826 25.60773 moveto 128.03738 110.96599 L 0 setlinecap stroke  end �������QyL�color push gray 0Figure�UU14:�q�Tw���o�p�Gossible�placemen�ts�of��
�b>

cmmi10�T����ٓ�Rcmr7�4��|s�	color pop����	color pop����W�N����,���E��2�,��X��[�,��and�Jthe�Ieast�v��9ertex�of��T�����7��*��.��The�latter�t�w�o�Jcannot�coincide�since����W�X�S�is�g�south�g�of��R�>Q��and�w��9est�of��P�H�Q��extended,�|�while�the�east�v�ertex�g�of��T�����7�����W�w��9ould���b�A�e�(in���either�p�ossible�orien��9tation�of��T�����7��*��)�east���of��P�H�Q��extended.���Since����W�T�����7���M�cannot�Χb�A�e�Φplaced�east�of��E��2Q�,���the�third�side�of�the�triangle�m��9ust�b�A�e����W�E��2Q�
��extended.��Hence�triangle��T�����6��5t�do�A�es�not�ha��9v�e�
�its�
�righ�t�angle�at��Q�
��(or�it����Ww��9ould�	extend�	east�of��E��2Q��extended).�YSince��R�>Q����=��b�,��triangle�	�T�����6��3��has�angle����W��
T�at��)�X��[�.���Hence�it��(has�either�angle������at��Q�.���Then�the�angle�sum�at��Q��is����W3��Yd�+�W����o�=�IC2���+���R�=�2�IC�<�IB��.�f�No��9w��lw�e��kha�v�e�a�six-triangle��kconguration�with����Wfour���v��9ertices����N����,��P�E��2�,��Q�Q�,�and��X��[�,�and��N���R�>�,��N�E��2�,�and����QE��are���sides�of�the�nal����Wtriangle.��But����T�����7����cannot�b�A�e�placed���south�of��X��[Q��so�as�to�create�a�triangle,����Wsince�X�to�do�X�so�it�w��9ould�need�an�obtuse�angle�at��X��[�.���This�con��9tradicts�our����Wassumption�&"that�&!�R�>W�o�is�a�side�of�the�nal�triangle.�N�Therefore��R�>W�o�is�not����Wa��oside�of��pthe�nal�triangle,��and�w��9e�are�bac��9k�to�where�only��T�����4����has�b�A�een����Wplaced���(along����W�H�Q��with�angle����I�at��Q�),���as�sho��9wn�in�the�rst�part�of�Figure����W15.����d��src:2621 SevenTriangles.texSince���R�>Q��is�parallel�to��P�H�E��2�,����R�Q��is�not�a�side�of�the�nal�triangle,���and����Ww��9e���m�ust�place����T�����5���N�south�of��R�>Q�.��7If�its�righ�t�angle���is�placed�at��R�>�,��1then�since����W�R�>W����is��not��a�side�of��AB�r�C����,���w��9e�will�require�b�A�oth��T�����6�����and��T�����7����w��9est�of��S��W����W�extended,�GWsince�
#w��9e�are�
$not�allo�w�ed�to�create�
$another�non-strict�v�ertex.����WThat�� will�lea��9v�e�� v�ertices�at���P�H��,��+�E��2�,��*�Q�,�and�p�A�oin��9ts�w�est.�
^Hence��T�����5����cannot�b�A�e����Wplaced�^with�^its�righ��9t�angle�at��R�>�.���But�then�the�righ��9t�angle�of��T�����5�����is�at��Q�.����WThis�5Wcreates�a�conca��9v�e�5Wv�ertex�at��Q��with�exterior�angle�greater�than���R�=�2,����Wso�[�t��9w�o�[�more�triangles�are�required�at��Q�,��lea��9ving�none�to�place�w��9est�of��R�>W�H��,����Wwhere��w��9e�need��one�since��R�>W����is�not�a�side�of��AB�r�C����.��|This�con��9tradiction����Wsho��9ws�Tthat�placing��T�����4��?��along��W�H�Q��with�angle����at��Q��fails.����d��src:2629 SevenTriangles.texHence��w��9e��m�ust�place���T�����4��3D�south�of��W�H�Q��with�angle������at��Q�,�(as�sho��9wn�in����Wthe�U�second�U�part�of�Figure�15.��Let��R�e��b�A�e�the�third�v��9ertex�of��T�����4��*��.��Supp�A�ose,����Wfor��Rpro�A�of�b��9y��Scon�tradiction,��that��R�W�H�P�D5�is�not�a�side�of�triangle��AB�r�C����.��Then����Ww��9e��[m�ust�add��\�T�����5���with�v�ertex����"�at��\�P�H��,�\side��c��along��W�H�P��,�]third�v��9ertex��N����W�on�>Y�P�H�E����extended,�H�with�>Zangle��W�N���P��=��%��R�=�2.���The�total�>Zangle�at��W��=�is�no��9w����W3���k�+����	��=�����R�=�2��+�2����.��T��:�o��
eliminate�the��v��9ertex�at��W����(making�a�straigh��9t�angle����Wat����W�H��)�w��9e�w�ould�need�an�angle�of����d��e��(��R�=�2�e�+�2����)�l�=�l���=�2���e��2����,��fwhic��9h���is����Wab�A�out���90��V���2����26�:�5���=�37���degrees,��more�than�����,��but�less�than�b�A�oth�2�����and����W��x,�,�9Cand�2hence�imp�A�ossible�2to�supply��:�.�r�P��9ossibly��W�z��migh�t�b�A�e�2eliminated�as�a�����>�color push gray 0�����44��Y�	color pop����-ݘ�y�������>�color push gray 0�Y�	color pop���?������W�v��9ertex�]�of�]��AB�r�C���b�y�b�A�ecoming�]�an�in��9ternal�v�ertex.��CThen�]�the�total�angle�at��W����W�w��9ould��)ha�v�e�to��*b�A�e�made�equal�to�2��R��.�
After�placing��T�����5�����it�is���=�2�c�+�c�2����,��2so�w��9e����Ww��9ould�q�need�q�2��G��Ɲ�(��R�=�2�+�2����)���=�3��=�4�Ɲ��ƞ�2��sM�more,��Jwhic��9h�q�is�q�ab�A�out�217�degrees.����WEv��9en��&if�w�e�used�the�righ�t�angles�of��T�����6�����and��T�����7���w��9e�could�not�mak�e�it.��aHence����W�W����is���denitely�a�v��9ertex�of���triangle��AB�r�C�V��(under�the�assumption�that��W�H�P����W�is���not���a�side.)��This�is,��3ho��9w�ev�er,��2imp�A�ossible,�since���t�w�o���of�the�v��9ertices�m�ust����Wb�A�e��pon�line��P�H�E�%��extended,��kand��o�W��S�cannot�b�e�the�southernmost�v��9ertex,��ksince����Wfor��example��p�A�oin��9t��Q��lies�farther�to�the�south�than��W�H��.�RThis�con��9tradiction����Wsho��9ws�	ythat�	x�W�H�P�R\�is,�F�in�fact,�F�one�of�the�sides�of�the�nal�triangle��AB�r�C����,����Walong�Twith��P�H�E��2�.����d��src:2643 SevenTriangles.texTh��9us����P���is���one�of�the�v�ertices�of��AB�r�C����,���and�the�other�t�w�o���lie�on��P�H�E����W�(extended)��and��P�H�W�1|�(extended).��<�W��cannot��b�A�e�a�v��9ertex��of��AB�r�C����,�isince����Wthen����W�Gr�w��9ould���b�A�e�the�southernmost�v�ertex�of����AB�r�C����,�8�but��R���lies���farther����Wsouth�than��W�H��.���Hence�the�v��9ertex�~at��W�Pb�m�ust�b�A�e�~eliminated�b�y�placing����Wmore�Btriangles�Cwith�a�v��9ertex�at��W�H��.�+;The�angle�sum�at��W�c%�(from�the�three����Wtriangles��already��
there)�is�3����,���ab�A�out�79�:�5�degrees.�T��:�o�reac��9h�an�angle�sum����Wof��S����at��W�H��,��w��9e��Rcould�place�three�triangles�with�angle���>�at��W�H��,��but�that����Ww��9ould��
use��all�sev�en�triangles��and�still�lea�v�e��v�ertices�at��P�H��,��|�E��2�,��{and���Q�,�as����Ww��9ell��yas�somewhere�south�w�est��xof��W��\�on��P�H�W��extended,��so�no�triangle�w��9ould����Wb�A�e�G4formed.��A��9t�least�t�w�o�m�ust�G3b�A�e�used�since�3��1?�+�/w��R�=�2����<���.��The�G4p�A�ossible����Wangle�Tsums�resulting�from�placing�t��9w�o�Tmore�angles�at��W�^7�are�among��������ܐ5�������k�<�����������������ߴ�4����+�8�������k��=��������3����+�8��R�=�2����<�����������o�4����+�8��R�=�2������k��>�����������������?��3����+�82�������k��=����������`��+�8����>���������������3����+�8���d�+���R�=�2������k�=��������2����+�8���p>���������W�src:2658 SevenTriangles.tex�None� Uof�these� Tp�A�ossibilities�w��9ould�succeed�to�eliminate�v��9ertex��W�H��.�=rThis����Wnal�Tcon��9tradiction�completes�the�pro�A�of�of�the�theorem.������W�References��阍���W�color push gray 0����[1]�	color pop���jc��src:2665 SevenTriangles.texBeeson,��M.,��Tiling��Sa�T��:�riangle�with�Congruen��9t�T�riangles,��to�appp�A�ear.������W�color push gray 0���[2]�	color pop���jc��src:2668 SevenTriangles.texBolt��9y�anskii,�^9V.�O�G.,�^8and�Goh��9b�A�erg,�I.�T.�O��The���de��c�omp�osition���of�gur�es����jc�into�h5smal�x�ler�h4p��arts�,�8�translated�1�and�adapted�1�from�the�Russian�edition����jc�b��9y�THenry�Christoers�and�Thomas�P��:�.�Branson.�QA167�.B6513������W�color push gray 0���[3]�	color pop���jc��src:2671 SevenTriangles.texBolt��9y�anskii,�TV.��G.���Equivalent�T�and�Equide��c�omp�osable�T�Figur�es�,�T��:�rans-����jc�lated�x0and�x/adapted�from�the�1st�Russian�ed.�(1956)�b��9y�Alfred�K.�Henn����jc�and�TCharles�E.�W��:�atts.�D.�C.�Heath�(1963).�QA447�.B573������W�color push gray 0���[4]�	color pop���jc��src:2674 SevenTriangles.texGoldb�A�erg,��M.,�and�qStew��9art,�B.�M.,�A�p�dissection�problem�for�sets�of����jc�p�A�olygons,�T�A�Îmer.�N<Math.�Monthly��71��(1964)�1077{1095.������W�color push gray 0���[5]�	color pop���jc��src:2677 SevenTriangles.texGolom��9b,�"BSolomon��W.,�Replicating�gures��in�the�plane,�D��The�W�Mathe-����jc�matic��al�N<Gazette�48�T�No.�366.�(Dec.,�1964),�pp.�403-412.�����>�color push gray 0�����45��Y�	color pop�����k���;�y�-��j��		cmti9���N�ffcmbx12�X�Qffcmr12�����		cmsy9�5��"		cmmi9�t�:		cmbx9�o���		cmr9�;�cmmi6��Aa�cmr6�X�Qcmr12�D��tG�G�cmr17�
�b>

cmmi10�K�`y

cmr10�ٓ�Rcmr7�w�����

Sindbad File Manager Version 1.0, Coded By Sindbad EG ~ The Terrorists