document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1082718>Question 1082718</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Triangle ABC with vertices (-J,0) (0,K) (L,0) and midpoints M,N,and O of the line segments Ab, BC, and AC.<BR>\n" );
document.write( "Prove the perimeter of triangle MNO is one half the perimeter 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 #696785 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=696785\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\"><BR>\n" );
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document.write( "Coordinates of midpoints (M, N, and O) derived from the midpoint formulas.  Verification left as an exercise for the student.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ AB\ =\ \sqrt{(-J\ -\ 0)^2\ +\ (0\ -\ K)^2}\ =\ \sqrt{J^2\ +\ K^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ BC\ =\ \sqrt{(0\ -\ L)^2\ +\ (K\ -\ 0)^2}\ =\ \sqrt{L^2\ +\ K^2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ AC\ =\ \sqrt{(-J\ -\ L)^2\ +\ (K\ -\ K)^2}\ =\ \sqrt{(J\ +\ L)^2}\ =\ J\ +\ L\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ MO\ =\ \sqrt{\left(-\frac{J}{2}\ -\ \frac{-J\ +\ L}{2}\right)^2\ +\ \left(\frac{K}{2}\ -\ 0\right)^2}\ =\ \sqrt{J^2\ +\ K^2}\ =\ \sqrt{\left(\frac{L}{2}\right)^2\ +\ \left(\frac{K}{2}\right)^2}\ =\ \frac{\sqrt{L^2\ +\ K^2}}{2}\ =\ \frac{BC}{2}\">\r<BR>\n" );
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document.write( "Similarly,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ NO\ =\ \frac{AB}{2}\">\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ MN\ =\ \frac{AC}{2}\">\r<BR>\n" );
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document.write( "But the perimeter of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large ABC\"> is the sum <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_{ABC}\ =\ AB\ +\ BC\ +\ AC\">\r<BR>\n" );
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document.write( "And the perimeter of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large MNO\"> is the sum <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P_{MNO}\ =\ MN\ +\ NO\ +\ MO\"> and:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P_{MNO}\ =\ \frac{AC}{2}\ +\ \frac{AB}{2}\ +\ \frac{BC}{2}\ =\ \frac{P_{ABC}}{2}\">\r<BR>\n" );
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document.write( "Q.E.D\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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