document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201144>Question 1201144</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given square ABCD, let P and Q be the points outside the square that make triangles<BR>\n" );
document.write( "CDP and BCQ equilateral. Prove that triangle AP Q is also equilateral. </SPAN>\n" );
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document.write( "Start with square ABCD. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28300%2C300%2C-4%2C4%2C-4%2C4%2C%0D%0A%0D%0Aline%280.85%2C0%2C1.15%2C0%29%2Cline%28-1.15%2C0%2C-0.85%2C0%29%2Cline%280%2C0.85%2C0%2C1.15%29%2Cline%280%2C-1.15%2C0%2C-0.85%29%2C%0D%0A%0D%0Aline%28-1%2C-1%2C1%2C-1%29%2Cline%281%2C-1%2C1%2C1%29%2Cline%281%2C1%2C-1%2C1%29%2Cline%28-1%2C1%2C-1%2C-1%29%2C%0D%0A%0D%0Acircle%28-1%2C-1%2C0.03%29%2Ccircle%28-1%2C-1%2C0.06%29%2Ccircle%28-1%2C-1%2C0.09%29%2Ccircle%281%2C-1%2C0.03%29%2Ccircle%281%2C-1%2C0.06%29%2Ccircle%281%2C-1%2C0.09%29%2Ccircle%281%2C1%2C0.03%29%2Ccircle%281%2C1%2C0.06%29%2Ccircle%281%2C1%2C0.09%29%2Ccircle%28-1%2C1%2C0.03%29%2Ccircle%28-1%2C1%2C0.06%29%2Ccircle%28-1%2C1%2C0.09%29%2C%0D%0A%0D%0Alocate%28-1%2C-1.1%2C%22A%22%29%2Clocate%281%2C-1.1%2C%22B%22%29%2Clocate%281%2C1.45%2C%22C%22%29%2Clocate%28-1%2C1.45%2C%22D%22%29%2C%0D%0A%0D%0Alocate%28-4%2C-3%2Cmatrix%281%2C4%2CDiagram%2Cis%2Cto%2Cscale%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Form equilateral triangles CDP and BCQ, such that the new points are outside the square.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28300%2C300%2C-4%2C4%2C-4%2C4%2C%0D%0A%0D%0Aline%280.85%2C0%2C1.15%2C0%29%2Cline%28-1.15%2C0%2C-0.85%2C0%29%2Cline%280%2C0.85%2C0%2C1.15%29%2Cline%280%2C-1.15%2C0%2C-0.85%29%2C%0D%0A%0D%0Aline%280.63%2C1.94%2C0.37%2C1.79%29%2Cline%28-0.63%2C1.94%2C-0.37%2C1.79%29%2Cline%281.94%2C-0.63%2C1.79%2C-0.37%29%2Cline%281.94%2C0.63%2C1.79%2C0.37%29%2C%0D%0A%0D%0Aline%28-1%2C-1%2C1%2C-1%29%2Cline%281%2C-1%2C1%2C1%29%2Cline%281%2C1%2C-1%2C1%29%2Cline%28-1%2C1%2C-1%2C-1%29%2C%0D%0A%0D%0Acircle%28-1%2C-1%2C0.03%29%2Ccircle%28-1%2C-1%2C0.06%29%2Ccircle%28-1%2C-1%2C0.09%29%2Ccircle%281%2C-1%2C0.03%29%2Ccircle%281%2C-1%2C0.06%29%2Ccircle%281%2C-1%2C0.09%29%2Ccircle%281%2C1%2C0.03%29%2Ccircle%281%2C1%2C0.06%29%2Ccircle%281%2C1%2C0.09%29%2Ccircle%28-1%2C1%2C0.03%29%2Ccircle%28-1%2C1%2C0.06%29%2Ccircle%28-1%2C1%2C0.09%29%2C%0D%0A%0D%0Alocate%28-1%2C-1.1%2C%22A%22%29%2Clocate%281%2C-1.1%2C%22B%22%29%2Clocate%281%2C1.45%2C%22C%22%29%2Clocate%28-1.2%2C1.45%2C%22D%22%29%2C%0D%0A%0D%0Aline%281%2C1%2C-1%2C1%29%2Cline%28-1%2C1%2C0%2C2.73%29%2Cline%280%2C2.73%2C1%2C1%29%2C%0D%0Aline%281%2C-1%2C1%2C1%29%2Cline%281%2C1%2C2.73%2C0%29%2Cline%282.73%2C0%2C1%2C-1%29%2C%0D%0A%0D%0Acircle%281%2C1%2C0.03865%29%2Ccircle%281%2C1%2C0.0773%29%2Ccircle%281%2C1%2C0.11595%29%2Ccircle%28-1%2C1%2C0.03865%29%2Ccircle%28-1%2C1%2C0.0773%29%2Ccircle%28-1%2C1%2C0.11595%29%2Ccircle%280%2C2.73%2C0.03865%29%2Ccircle%280%2C2.73%2C0.0773%29%2Ccircle%280%2C2.73%2C0.11595%29%2Ccircle%281%2C-1%2C0.03865%29%2Ccircle%281%2C-1%2C0.0773%29%2Ccircle%281%2C-1%2C0.11595%29%2Ccircle%281%2C1%2C0.03865%29%2Ccircle%281%2C1%2C0.0773%29%2Ccircle%281%2C1%2C0.11595%29%2Ccircle%282.73%2C0%2C0.03865%29%2Ccircle%282.73%2C0%2C0.0773%29%2Ccircle%282.73%2C0%2C0.11595%29%2C%0D%0A%0D%0Alocate%280%2C3.23%2C%22P%22%29%2Clocate%282.73%2C-0.25%2C%22Q%22%29%2C%0D%0A%0D%0Alocate%28-4%2C-3%2Cmatrix%281%2C4%2CDiagram%2Cis%2Cto%2Cscale%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then form triangle APQ shown in red.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28300%2C300%2C-4%2C4%2C-4%2C4%2C%0D%0A%0D%0Aline%280.85%2C0%2C1.15%2C0%29%2Cline%28-1.15%2C0%2C-0.85%2C0%29%2Cline%280%2C0.85%2C0%2C1.15%29%2Cline%280%2C-1.15%2C0%2C-0.85%29%2C%0D%0A%0D%0Aline%280.63%2C1.94%2C0.37%2C1.79%29%2Cline%28-0.63%2C1.94%2C-0.37%2C1.79%29%2Cline%281.94%2C-0.63%2C1.79%2C-0.37%29%2Cline%281.94%2C0.63%2C1.79%2C0.37%29%2C%0D%0A%0D%0Aline%28-1%2C-1%2C1%2C-1%29%2Cline%281%2C-1%2C1%2C1%29%2Cline%281%2C1%2C-1%2C1%29%2Cline%28-1%2C1%2C-1%2C-1%29%2C%0D%0A%0D%0Acircle%28-1%2C-1%2C0.03%29%2Ccircle%28-1%2C-1%2C0.06%29%2Ccircle%28-1%2C-1%2C0.09%29%2Ccircle%281%2C-1%2C0.03%29%2Ccircle%281%2C-1%2C0.06%29%2Ccircle%281%2C-1%2C0.09%29%2Ccircle%281%2C1%2C0.03%29%2Ccircle%281%2C1%2C0.06%29%2Ccircle%281%2C1%2C0.09%29%2Ccircle%28-1%2C1%2C0.03%29%2Ccircle%28-1%2C1%2C0.06%29%2Ccircle%28-1%2C1%2C0.09%29%2C%0D%0A%0D%0Alocate%28-1%2C-1.1%2C%22A%22%29%2Clocate%281%2C-1.1%2C%22B%22%29%2Clocate%281%2C1.45%2C%22C%22%29%2Clocate%28-1.2%2C1.45%2C%22D%22%29%2C%0D%0A%0D%0Aline%281%2C1%2C-1%2C1%29%2Cline%28-1%2C1%2C0%2C2.73%29%2Cline%280%2C2.73%2C1%2C1%29%2C%0D%0Aline%281%2C-1%2C1%2C1%29%2Cline%281%2C1%2C2.73%2C0%29%2Cline%282.73%2C0%2C1%2C-1%29%2C%0D%0A%0D%0Acircle%281%2C1%2C0.03865%29%2Ccircle%281%2C1%2C0.0773%29%2Ccircle%281%2C1%2C0.11595%29%2Ccircle%28-1%2C1%2C0.03865%29%2Ccircle%28-1%2C1%2C0.0773%29%2Ccircle%28-1%2C1%2C0.11595%29%2Ccircle%280%2C2.73%2C0.03865%29%2Ccircle%280%2C2.73%2C0.0773%29%2Ccircle%280%2C2.73%2C0.11595%29%2Ccircle%281%2C-1%2C0.03865%29%2Ccircle%281%2C-1%2C0.0773%29%2Ccircle%281%2C-1%2C0.11595%29%2Ccircle%281%2C1%2C0.03865%29%2Ccircle%281%2C1%2C0.0773%29%2Ccircle%281%2C1%2C0.11595%29%2Ccircle%282.73%2C0%2C0.03865%29%2Ccircle%282.73%2C0%2C0.0773%29%2Ccircle%282.73%2C0%2C0.11595%29%2C%0D%0A%0D%0Alocate%280%2C3.23%2C%22P%22%29%2Clocate%282.73%2C-0.25%2C%22Q%22%29%2C%0D%0A%0D%0Ared%28%0D%0Aline%28-1%2C-1%2C0%2C2.73%29%2Cline%280%2C2.73%2C2.73%2C0%29%2Cline%282.73%2C0%2C-1%2C-1%29%0D%0A%29%2C%0D%0A%0D%0Alocate%28-4%2C-3%2Cmatrix%281%2C4%2CDiagram%2Cis%2Cto%2Cscale%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The goal is to prove that triangle APQ is equilateral.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For now let's focus on triangle ABQ.<BR>\n" );
document.write( "BQ = BC (since BCD is equilateral)<BR>\n" );
document.write( "AB = BC (since ABCD is a square)<BR>\n" );
document.write( "AB = BQ (by the transitive property)\r<BR>\n" );
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document.write( "The AB = BQ statement then tells us triangle ABQ is isosceles.<BR>\n" );
document.write( "The vertex angle is 90+60 = 150 degrees.<BR>\n" );
document.write( "The base angles are (180-150)/2 = 30/2 = 15 degrees each.\r<BR>\n" );
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document.write( "Through similar logic, you'll find that triangles ADP and PCQ are congruent to isosceles triangle ABQ.<BR>\n" );
document.write( "All isosceles triangles mentioned have the same base length<BR>\n" );
document.write( "AQ = AP = PQ\r<BR>\n" );
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document.write( "Therefore, triangle APQ is equilateral.<BR>\n" );
document.write( "</font><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );