document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=510285>Question 510285</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help writing a coordinate proof for the statement: The segments joining the vertices to the midpoints of the legs of an isosceles triangle are congruent. I cannot figure out how to start it off as to what would be my Given and what i'm trying to prove. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #341805 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=341805\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To construct a coordinate proof, you will have to fix points first. Without loss of generality suppose that for isosceles triangle ABC, A is at (0,0), B is at (2a,0), and vertex C is at (a,b) (so that AC = BC).\r<BR>\n" );
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document.write( "Let D and E be the midpoints of AC and BC respectively. We want to prove that BD = AE. We can easily find the coordinates of D and E using the fact that they're midpoints:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE D = (\frac{a}{2}, \frac{b}{2})\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE E = (\frac{3a}{2}, \frac{b}{2})\">\r<BR>\n" );
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document.write( "Then the distance from B to D is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \sqrt{(\frac{3a}{2})^2 + (\frac{b}{2})^2}\"> (using the distance formula/Pythagorean theorem)\r<BR>\n" );
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document.write( "Similarly, the distance from A to E is the same amount. Hence BD = AE, and we are done.\r<BR>\n" );
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