document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=448086>Question 448086</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Could you please help me solve this problem?\r<BR>\n" );
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document.write( "Prove the Median Inequality: If M is the midpoint of side BC of triangle ABC, then AM is less than 1/2(AB+AC).\r<BR>\n" );
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document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #308474 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=308474\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Whoops, mixed up AM with BM. It occasionally happens when I try to solve a question without pen or paper. Anyway, I have come up with a new proof, which is similar to that of the other tutor's, but uses a different method of constructing it. Note that neither proof is necessarily more \"correct\" than the other.\r<BR>\n" );
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document.write( "Start with the original diagram (I have used the other tutor's as a reference):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C1000%2F3%2C-6%2C6%2C-5%2C5%2C+triangle%281%2C4%2C-5%2C0%2C5%2C0%29%2C+%0D%0Agreen%28line%281%2C4%2C0%2C0%29%29%2C+locate%28-5%2C0%2CB%29%2C+locate%285%2C0%2CC%29%2C+locate%281%2C4.5%2CA%29%2C%0D%0Alocate%280%2C0%2CM%29%0D%0A+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can extend CA past A, and draw a line through B parallel to AM, and call their intersection E:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C1000%2F3%2C-6%2C6%2C-5%2C5%2C+triangle%281%2C0%2C-5%2C-4%2C5%2C-4%29%2C+%0D%0Agreen%28line%281%2C0%2C0%2C-4%29%29%2C+locate%28-5%2C-4%2CB%29%2C+locate%285%2C-4%2CC%29%2C+locate%281%2C0.5%2CA%29%2C%0D%0Alocate%280%2C-4%2CM%29%2C%0D%0A%0D%0A%0D%0Aline%281%2C+0%2C+-3%2C+4%29%2C+line%28-5%2C+-4%2C+-3%2C+4%29%2C+locate%28-3%2C+4.5%2C+E%29%0D%0A+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "By AAA similarity, we can show that triangles EBC and AMC are similar, with a ratio of 2:1 (since BC = CM). Hence, we can establish the following ratios:\r<BR>\n" );
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document.write( "EB = 2AM<BR>\n" );
document.write( "AC = AE\r<BR>\n" );
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document.write( "From the triangle inequality, EB < AB + AE. Since EB = 2AM and AC = AE, we obtain 2AM < AB + AC --> AM < (AB + AC)/2. </SPAN>\n" );
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