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    <h2>Tarski Formalization Project Archives</h2>
    <p></p>
    <h3>Reflection in a point, midpoint of the base of an isosceles triangle, and the Krippenlemma</h3>
    <p></p>
    <p>
    The reflection of $a$ in $c$ is the unique point $x$ such that $T(a,c,x)$ and $ac = xc$.  It is
    written $x=s(c,a)$.  This chapter presents results from Gupta's thesis that are used in Chapter 8
    to help prove his famous results about the existence of perpendiculars.  Remember that in this chapter,
    we are not using the parallel axiom, any form of continuity (including line-circle and circle-circle continuity),
    and not even the upper dimension axiom.
     </p>
     
     <p>  The predicate $M(a,m,b)$ means that $am = mb$; $m$  (if it exists) is unique.  Until we prove that it 
     does exist, we can't introduce the notation $midpoint(a,b)$; but after proving that the base of an isosceles
     triangle $abc$, with $ac=bc$, exists, we can introduce $isomidpoint(a,b,c)$ for the midpoint of $ab$. 
     </p>
      
<?php
$Chapter="7";
$descriptions["Satz7.2"] = "\$M(a,m,b) \\rightarrow M(b,m,a)\$";
$descriptions["Satz7.3a"] = "\$M(a,m,b) \\rightarrow m=a \$";
$descriptions["Satz7.3b"] = "\$m=a \\rightarrow M(a,m,b)\$";
$descriptions["Satz7.4a"] = "\$M(p,a,s(a,p))\$";
$descriptions["Satz7.4b"] = "Uniqueness of reflection in a point. <br>" .
                            "\$ M(p,a,r) \\land M(p,a,q) \\rightarrow r=q\$";
$descriptions["Satz7.6"] = "\$M(p,a,q) \\rightarrow q = s(a,p)\$";
$descriptions["Satz7.7"] = "\$s(a,s(a,p)) = p\$ (reflection is an involution). <br> In the book this is repeated as Satz 7.12.";
$descriptions["Satz7.8"] = "\$s(a,p)= r \\land s(a,q) = r \\rightarrow p = q\$";
$descriptions["Satz7.9"] = "\$s(a,p)=  s(a,q)   \\rightarrow p = q\$";
$descriptions["Satz7.10a"] = "\$s(a,p)=p \\rightarrow p=a\$";
$descriptions["Satz7.10b"] = "\$ s(a,p) \\neq p \\rightarrow p \\neq a\$";
$descriptions["Satz7.13"] = "\$ E(p,q,s(a,p),s(a,q))\$";
$descriptions["Satz7.15a"] = "Reflection preserves betweenness.";
$descriptions["Satz7.15b"] = "Converse of Satz 7.15a";
$descriptions["Satz7.16a"] = "Reflection preserves congruence ";
$descriptions["Satz7.16b"] = "Converse of Satz 7.16a";
$descriptions["Satz7.17"] = "Uniqueness of midpoint";
$descriptions["Satz7.18"] = "\$ s(a,p)=s(b,p)\\rightarrow a=b\$";
$descriptions["Satz7.19"] = "\$ s(a,s(b,p)) = s(b,s(a,p)) \\rightarrow a=b \$";
$descriptions["Satz7.20"] = "\$Col(a,m,b) \\land E(m,a,m,b) \\rightarrow a=b \\lor M(a,m,b)\$";
$descriptions["Satz7.21"] = "In a quadrilateral with opposite sides equal, the diagonals bisect each other." .
        "(It is assumed that the diagonals meet, which is necessary since no dimension axiom is used.)" .
         " This is one of Quaife's challenge problems.";
$descriptions["Satz7.22a"] = "Gupta's  Krippenlemma  (an important lemma for Chapter 8)," .
            "proved with additional assumption  \$ca_1 \le ca_2\$.   The
              Krippenlemma is derived by two applications of this theorem,
              as in the book, but in the book this case is not stated separately.";
$descriptions["Satz7.22b"] = "Krippenlemma, stated using abbreviation KF.";
$descriptions["Satz7.22"] = "Krippenlemma, stated as in the book (without using the abbreviation KF).";
$descriptions["Satz7.25"] = "The base of an isosceles triangle has a midpoint.";


include("DisplayResults.php");
?>

<!--
      v
       <tr>
        <td><a href="InputFiles/Satz7.21.in">Satz 7.21</a></td>
        <td><a href="Proofs/Satz7.21.prf">Satz7.21.prf</a><br>31 steps</td>
        <td> In a quadrilateral with opposite sides equal <br> the diagonals bisect each other. 
        (It is assumed that the diagonals meet, which is necessary since no dimension axiom is used.)
        <br> This is one of Quaife's challenge problems.</td>
      </tr>
      <tr>
        <td><a href="InputFiles/Satz7.22a.in">Satz 7.22a</a></td>
        <td><a href="Proofs/Satz7.22a.prf">Satz7.22a.prf</a>
            <br> 96 steps</td>
        <td> Gupta's  Krippenlemma <br> (an important lemma for Chapter 8)
            <br> Proved with additional assumption  ca1 &le; ca2.   The
              Krippenlemma is derived by two applications of this theorem,
              as in the book, but in the book this case is not stated separately.
        </td>
      </tr>

         <tr>
        <td><a href="InputFiles/Satz7.22.in">Satz 7.22</a></td>
        <td><a href="Proofs/Satz7.22.prf">Satz7.22.prf</a>
            <br> 12 steps</td>
        <td> Gupta's  Krippenlemma <br> (an important lemma for Chapter 8)</td>
      </tr>
          <tr>
        <td><a href="InputFiles/Satz7.25.in">Satz 7.25</a></td>
        <td><a href="Proofs/Satz7.25.prf">Satz7.25.prf</a><br>123 steps</td>
        <td> base of an isosceles triangle has a midpoint</td>
      </tr>
      
    </table>
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