Sindbad~EG File Manager

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    <h2>Tarski Formalization Project Archives</h2>
        <p>The posted input files are now all mechanically generated from a master list of theorems.
    For more information about our methodology see the <a href="http://www.michaelbeeson.com/research/FormalTarski/index.php"> top page of this project</a>.
  </p>
    <h3>Congruence of right triangles; triangle construction and uniqueness</h3>
    <p>The most important theorem in this chapter is Satz 10.12, which says that two right triangles 
    with pairwise congruent legs also have congruent hypotenuses.  </p> 
    <p>  This is not an immediate consequence of the 5-segment axiom;
             the 5-segment axiom together with this theorem implies SAS, but the 
             5-segment axiom alone does not immediately imply SAS. </p>
    <p> Euclid took it as a postulate that all right angles are congruent, which is essentially the 
    content of Satz 10.12, although angle congruence is not defined until Chapter 11.   
    </p>
    
    <p> Satz 10.12 is proved in three stages:
    first we reduce to the case where the two triangles have their right-angled vertex in common 
    (by reflection in the midpoint of the line joining those vertices).  Then we reduce to the case 
    where they have one leg in common.  To that we need a "rotation".  Rotation through a given angle 
    can be done by reflecting in the bisector of that angle.  Since we can find the midpoint of a line,
    by Gupta's midpoint construction, we can bisect an angle.  So,  the chapter begins with the definition
    and properties of reflection in a line.  The only difficult property is that reflection preserves betweenness 
    (Satz 10.10). 
    </p>
    <p>Once Satz 10.12 is in hand, and Gupta's dropped perpendiculars are available, it is an easy 
    matter to verify Hilbert's axiom about triangle construction and uniqueness (Satz 10.16a and Satz 10.16b).
    </p>
    
    <table width="700" border="1">
      <tr>
        <th width="114" scope="col">Input File</th>
        <th width="123" scope="col">Proof</th>
        <th width="365" scope="col">Commentary</th>
      </tr>
       <tr>
        <td><a href="InputFiles/Satz10.2a.in">Satz 10.2a</a></td>
        <td><a href="Proofs/Satz10.2a.prf">Satz10.2a.prf</a><br>45 steps</td>
        <td>existence of reflect(a,b,p), the reflection of p in Line(a,b)  (provided a!=b)</td>
      </tr>
      <tr>
        <td><a href="InputFiles/Satz10.2b.in">Satz 10.2b</a></td>
        <td><a href="Proofs/Satz10.2b.prf">Satz10.2b.prf</a><br>not yet proved in one file</td>
        <td> Uniqueness of the reflection of a point in a line. 
        <br> So far we can prove this in two cases but can't combine the cases.
        </td>
      </tr>
       <tr>
        <td><a href="InputFiles/Satz10.4.in">Satz 10.4</a></td>
        <td><a href="Proofs/Satz10.4.prf">Satz10.4.prf</a> <br>19 steps</td>
        <td> reflect(a,b,p) = q implies reflect(a,b,q) = p  (provided a!=b)</td>
      </tr>
          <tr>
        <td><a href="InputFiles/Satz10.5.in">Satz 10.5</a></td>
        <td><a href="Proofs/Satz10.5.prf">Satz10.5.prf</a><br>1 step</td>
        <td> reflect(a,b,reflect(a,b,p)) = p  (provided a!=b) </td>
      </tr>
     </tr>
     <tr>
        <td><a href="InputFiles/Satz10.6.in">Satz 10.6</a></td>
        <td><a href="Proofs/Satz10.6.prf">Satz10.6.prf</a><br> 4 steps</td>
        <td> There exists exactly one x such that reflect(a,b,x) = p, <br> namely x = reflect(a,b,reflect(a,b,p)). </td>
      </tr>
      <tr>
        <td><a href="InputFiles/Satz10.7.in">Satz 10.7</a></td>
        <td><a href="Proofs/Satz10.7.prf">Satz10.7.prf</a><br>3 steps</td>
        <td> reflection in Line(a,b) is one-to-one: <br>
             reflect(a,b,p) = reflect(a,b,q) implies p=q.</td>
      </tr>
      <tr>
        <td><a href="InputFiles/Satz10.8a.in">Satz 10.8a</a></td>
        <td><a href="Proofs/Satz10.8a.prf">Satz10.8a.prf</a><br>3 steps</td>
        <td> reflect(a,b,p) = p implies Col(a,b,p) ) (provided a!=b)</td>
      </tr>
      <tr>
        <td><a href="InputFiles/Satz10.8b.in">Satz 10.8b</a></td>
        <td><a href="Proofs/Satz10.8b.prf">Satz10.8b.prf</a><br>1 step</td>
        <td> if Col(a,b,p) then reflect(a,b,p) = p  (provided a!=b)</td>
      </tr>
       <tr>
        <td><a href="InputFiles/Satz10.10a.in">Satz 10.10a</a></td>
        <td><a href="Proofs/Satz10.10a.prf">Satz10.10a.prf</a><br>33 steps
            
        </td>
        <td>   The case Col(a,b,p) of Satz 10.10. <br>
        We had to prove this case separately as a lemma.
        </td>
      </tr>
       <tr>
        <td><a href="InputFiles/Satz10.10.in">Satz 10.10</a></td>
        <td><a href="Proofs/Satz10.10.prf">Satz10.10.prf</a><br>
             86 steps
        </td>
        <td> Reflection preserves congruence: <br>
             E(p,q,reflect(a,b,p),reflect(a,b,q)) (provided a!=b) <br>
             Our proof uses the lemmas in Satz10.10a.in and Lemma9.3.in.
        </td>
      </tr>
      <tr>
        <td><a href="InputFiles/Satz10.12b.in">Satz 10.12b</a></td>
        <td><a href="Proofs/Satz10.12b.prf">Satz10.12b.prf</a><br>105 steps</td>
        <td> Two right triangles with congruent legs, a common vertex at the right triangle, 
           and one common leg, have congruent hypotenuses. <br>
        </td>
      </tr>
       <tr>
        <td><a href="InputFiles/Satz10.12c.in">Satz 10.12c</a></td>
        <td><a href="Proofs/Satz10.12c.prf">Satz10.12c.prf</a><br>11 steps</td>
        <td> Two right triangles with a common vertex at the right angle and congruent legs have congruent hypotenuses,
           if two equal legs are collinear on opposite sides of the vertex.  <br>
           This case is not treated separately in the book, but we had to do so.
        </td>
      </tr>
       <tr>
        <td><a href="InputFiles/Satz10.12a.in">Satz 10.12a</a></td>
        <td><a href="Proofs/Satz10.12a.prf">Satz10.12a.prf</a><br>62 steps</td>
        <td> Two right triangles with a common vertex at the right angle and congruent legs have congruent hypotenuses. <br>
        </td>
      </tr>
      <tr>
        <td><a href="InputFiles/Satz10.12.in">Satz 10.12</a></td>
        <td><a href="Proofs/Satz10.12.prf">Satz10.12.prf</a> <br>26 steps</td>
        <td> Two right triangles with congruent legs have congruent hypotenuses. <br>
          As in the book,
             we prove this theorem by reducing to 10.12a (common vertex), then to 10.12b (common leg), although
             in the book these theorems are not given different numbers. 
        </td>
      <tr>
        <td><a href="InputFiles/Satz10.14.in">Satz 10.14</a></td>
        <td><a href="Proofs/Satz10.14.prf">Satz10.14.prf</a> <br>10 steps</td>
        <td> if p1 = reflect(a,b,p) and p is not on Line(a,b), <br> then p and p1 are on opposite sides of Line(a,b)
        </td>
      </tr>
       <tr>
        <td><a href="InputFiles/Satz10.15.in">Satz 10.15</a></td>
        <td><a href="Proofs/Satz10.15.prf">Satz10.15.prf</a> <br>78 steps</td>
        <td>  if b!=c and Col(b,c,a) and not Col(b,c,q) then perp(b,c,p,a) and <br>
              p and q are on the same side of Line(b,c), <br>
             where p = erectsameside(b,c,a,q) is constructed in the proof.     
        </td>
      </tr>
      <tr>
        <td><a href="InputFiles/Satz10.16a.in">Satz 10.16a</a></td>
        <td><a href="Proofs/Satz10.16a.prf">Satz10.16a.prf</a><br>49 steps</td>
        <td> Triangle construction.   Given triangle abc and segment AB with ab=AB,
             and point p not on L = Line(A,B), there 
             exists point C on the same side of L as p such that triangle ABC is 
             congruent to triangle abc.
        </td>
      </tr>
        <tr>
        <td><a href="InputFiles/Satz10.16b.in">Satz 10.16b</a></td>
        <td><a href="Proofs/Satz10.16b.prf">Satz10.16b.prf</a> <br>60 steps</td>
        <td> Triangle uniqueness.   Given triangle abc and segment AB with ab=AB,
             and point p not on L = Line(A,B), there do not exist two different 
             points C on the same side of L as p such that triangle ABC is 
             congruent to triangle abc.
        </td>
      </tr>
    </table>
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