document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1155796>Question 1155796</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In triangle ABC, point P is on side BC such that PA = 13, PB = 14, PC = 4, and the circumcircle of triangles APB and APC have the same radius. Find the area of triangle ABC. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #778481 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=778481\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "In triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC ALIGN=MIDDLE  ALT=\"ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" >, point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > is on side <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC ALIGN=MIDDLE  ALT=\"BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > such that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+PA+=+13 ALIGN=MIDDLE  ALT=\"+PA+=+13\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PB+=+14 ALIGN=MIDDLE  ALT=\"PB+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PC+=+4 ALIGN=MIDDLE  ALT=\"PC+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "and the circumcircle of triangles <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=APB ALIGN=MIDDLE  ALT=\"APB\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=APC ALIGN=MIDDLE  ALT=\"APC\"   DISABLED_style= \"max-width:90%; height:auto;\" > have the same radius. Find the area of triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC ALIGN=MIDDLE  ALT=\"ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF=\"https://www.imageupload.net/image/a4e0s\"><IMG STYLE=\"max-width:80%;\" src=\"https://imagehost.imageupload.net/2020/04/11/triangle-ABC.th.png\" alt=\"triangle-ABC.png\" border=\"0\" /></a>\r<BR>\n" );
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document.write( "Note that  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AP ALIGN=MIDDLE  ALT=\"AP\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=chord ALIGN=MIDDLE  ALT=\"chord\"   DISABLED_style= \"max-width:90%; height:auto;\" > of both circles. Since both circles have the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=same ALIGN=MIDDLE  ALT=\"same\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=radius ALIGN=MIDDLE  ALT=\"radius\"   DISABLED_style= \"max-width:90%; height:auto;\" >, chord <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AP ALIGN=MIDDLE  ALT=\"AP\"   DISABLED_style= \"max-width:90%; height:auto;\" >  must subtend the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=same ALIGN=MIDDLE  ALT=\"same\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle, i.e. < <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABP ALIGN=MIDDLE  ALT=\"ABP\"   DISABLED_style= \"max-width:90%; height:auto;\" >= < <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ACP ALIGN=MIDDLE  ALT=\"ACP\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Thus, triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC ALIGN=MIDDLE  ALT=\"ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=isosceles ALIGN=MIDDLE  ALT=\"isosceles\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=AC ALIGN=MIDDLE  ALT=\"AB=AC\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Let<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+M ALIGN=MIDDLE  ALT=\"+M\"   DISABLED_style= \"max-width:90%; height:auto;\" > be the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=midpoint ALIGN=MIDDLE  ALT=\"midpoint\"   DISABLED_style= \"max-width:90%; height:auto;\" > of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC ALIGN=MIDDLE  ALT=\"BC\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We see that \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=BP%2B+PC=14%2B4=18 ALIGN=MIDDLE  ALT=\"BC=BP%2B+PC=14%2B4=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+BM=CM=BC%2F2=9 ALIGN=MIDDLE  ALT=\"+BM=CM=BC%2F2=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PM=MC-PC=9-4=5 ALIGN=MIDDLE  ALT=\"PM=MC-PC=9-4=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AM ALIGN=MIDDLE  ALT=\"AM\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=altitude ALIGN=MIDDLE  ALT=\"altitude\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can apply Pythagoras theorem to right triangle  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AMP ALIGN=MIDDLE  ALT=\"AMP\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AM ALIGN=MIDDLE  ALT=\"AM\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AM=sqrt%2813%5E2-5%5E2%29=sqrt%28169-25%29=sqrt%28144%29=12 ALIGN=MIDDLE  ALT=\"AM=sqrt%2813%5E2-5%5E2%29=sqrt%28169-25%29=sqrt%28144%29=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Hence, the area of triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ABC ALIGN=MIDDLE  ALT=\"ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" > is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=%281%2F2%29AM%2ABC ALIGN=MIDDLE  ALT=\"area=%281%2F2%29AM%2ABC\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=%281%2F2%2912%2A18 ALIGN=MIDDLE  ALT=\"area=%281%2F2%2912%2A18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=6%2A18 ALIGN=MIDDLE  ALT=\"area=6%2A18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=108 ALIGN=MIDDLE  ALT=\"area=108\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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