Given: ABCD is a square and XYZW is a figure inside the square
\n" );
document.write( "AX is congruent to BY which is congruent to CZ which is congruent to DW
\n" );
document.write( "BX is congruent to CY which is congruent to DZ which is congruent to AW
\n" );
document.write( "Prove that WXYZ is a square
\n" );
document.write( "\r\n" );
document.write( "I'll just outline the proof. You will have to write it\r\n" );
document.write( "up as a two-column proof.\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now these four outer triangles shown below\r\n" );
document.write( "are easily proved congruent by SSS:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Now by subtracting the sum of the measures of a\r\n" );
document.write( "pair of congruent angles in each corner of the \r\n" );
document.write( "big given square from 90° you get that these \r\n" );
document.write( "four angles at A, B, C, and D, shown below are \r\n" );
document.write( "all congruent:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Therefore the four triangles shown below around \r\n" );
document.write( "the center quadrilateral are congruent by SAS:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Therfore the center figure WXYZ is at least a \r\n" );
document.write( "rhombus, that is, all four sides are congruent,\r\n" );
document.write( "by \"corresponding parts of congruent triangles\".\r\n" );
document.write( "\r\n" );
document.write( "Now let's go back to the original figure:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Finally at each corner point of the inside figure,\r\n" );
document.write( "X, Y, Z, and W, there are 4 adjacent angles which have\r\n" );
document.write( "sum 360°. Three of the angles at each of those\r\n" );
document.write( "corner points X,Y,Z, and W, are congruent to three \r\n" );
document.write( "of the angles at each of the other corner point.\r\n" );
document.write( "\r\n" );
document.write( "So by subtracting the measures of those three angles \r\n" );
document.write( "from 360°, we get that each of the interior angles of the \r\n" );
document.write( "the inner quadrilateral WXYZ have the same measure.\r\n" );
document.write( "\r\n" );
document.write( "Now since the sum of the measures of the interior\r\n" );
document.write( "angles of any quadrilateral is (4-2)×180° or 360°,\r\n" );
document.write( "each of the corner points of the inside quadrilateral\r\n" );
document.write( "must be 360°÷4 or 90°. Thus it is a rhombus with\r\n" );
document.write( "four 90° interior angles, which is a square. \r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
