document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1100490>Question 1100490</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the area of an equilateral triangle if a point inside the triangle is at a distance of 3, 4 and 5 cm from the vertices? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #715124 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=715124\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br>This is the EXACT problem that, about 45 years ago, rekindled my interest in mathematical problem solving.<br><BR>\n" );
document.write( "A college friend of mine who had become a junior high school math teacher sent it to me, and over the course of a couple of years I made various attempts at solving it.  I know I finally came up with an answer after a huge amount of ugly mathematics; but that solution method is long lost.<br><BR>\n" );
document.write( "It is lost because, when I looked at the problem for the umpteenth time, I found a quick and simple solution that gives an exact answer.<br><BR>\n" );
document.write( "Here it is....<br><BR>\n" );
document.write( "Call the equilateral triangle ABC, with interior point P.  AP=3; BP=4, CP=5.<br><BR>\n" );
document.write( "Make a copy of triangle BPC.  Rotate that triangle 60 degrees about point B, so that the copy of side BC coincides with side BA.  Make P' the \"image\" of point P.<br><BR>\n" );
document.write( "Now consider quadrilateral BPAP'.  Draw diagonal PP'.<br><BR>\n" );
document.write( "Angle PBP' is 60 degrees; because it is still the sum of angles ABP and CBP.<br><BR>\n" );
document.write( "But BP and BP' are both 4 units long; that makes triangle BPP' an equilateral triangle, so PP' is 4 units long, and angle BPP' is 60 degrees.<br><BR>\n" );
document.write( "But now triangle APP' has side lengths 3, 4, and 5, so angle APP' is a right angle.<br><BR>\n" );
document.write( "And that makes angle BPA 150 degrees.<br><BR>\n" );
document.write( "And then the law of cosines gives us the exact length of the side s of triangle ABC; and from there we can find the area of triangle ABC.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2+=+3%5E2%2B4%5E2-2%2A3%2A4%2Acos%28150%29 ALIGN=MIDDLE  ALT=\"s%5E2+=+3%5E2%2B4%5E2-2%2A3%2A4%2Acos%28150%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2+=+25-24%2A%28-sqrt%283%29%2F2%29 ALIGN=MIDDLE  ALT=\"s%5E2+=+25-24%2A%28-sqrt%283%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2+=+25%2B12%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"s%5E2+=+25%2B12%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The area of an equilateral triangle with side length s is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2%2A%28sqrt%283%29%2F4%29 ALIGN=MIDDLE  ALT=\"s%5E2%2A%28sqrt%283%29%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so the area of the triangle is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2825%2B12%2Asqrt%283%29%29%2A%28sqrt%283%29%2F4%29+=+%2825%2Asqrt%283%29%2B36%29%2F4 ALIGN=MIDDLE  ALT=\"%2825%2B12%2Asqrt%283%29%29%2A%28sqrt%283%29%2F4%29+=+%2825%2Asqrt%283%29%2B36%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><BR>\n" );
document.write( "It is always very gratifying to find a simple solution to a problem that for so long had seemed extremely difficult....<br><BR>\n" );
document.write( "------------------------------------------------------------<br><BR>\n" );
document.write( "Note tutor ikleyn's technology solution gives the right answer for the length of the side of the triangle:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+sqrt%2825%2B12%2Asqrt%283%29%29+=+6.766432568 ALIGN=MIDDLE  ALT=\"s+=+sqrt%2825%2B12%2Asqrt%283%29%29+=+6.766432568\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "which confirms her statement that the side length is between 6.76 and 6.77. </SPAN>\n" );
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