document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1118483>Question 1118483</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I go about solving this math problem?\r<BR>\n" );
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document.write( "In a triangle, &#916;ABC the segment AM is the median to BC. Let D be the midpoint of AM and let N be the point on AC such that the points B, D, and N are on the same line. Find the ratio of DN to DB.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #734086 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=734086\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The answer is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+highlight%281%2F3%29+ ALIGN=MIDDLE  ALT=\"+highlight%281%2F3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <br>\r<BR>\n" );
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document.write( "This is admittedly a long procedure so there may very well be shorter ones.   I'm confident the answer is correct as I tried it with a sample triangle and it works.<br>\r<BR>\n" );
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document.write( "There are a lot of algebraic steps, so I will just outline the procedure. \r<BR>\n" );
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document.write( "Draw  the triangle ABC such that  A and B lie on the x-axis.  So triangle ABC has the following vertices:\r<BR>\n" );
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document.write( "A@ (0,0)<BR>\n" );
document.write( "B@ (Bx, 0)    <— notation is B sub x  = x coordinate of B, sorry it can look a little confusing<BR>\n" );
document.write( "C@ (Cx,Cy)    <br>\r<BR>\n" );
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document.write( "Point M is then at ((Bx+Cx)/2, Cy/2)<BR>\n" );
document.write( "Point D is at ((Bx+Cx)/4, Cy/4)\r<BR>\n" );
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document.write( "1.  Find the equation of line BD.  You will find it is  y = (Cy/(Cx-3Bx))x - (Bx*Cy)/(Cx-3Bx)<BR>\n" );
document.write( "2.  Find the equation of line AC.   You will find it is  y = (Cy/Cx)x <BR>\n" );
document.write( "3.  Set these two equations equal to each other to find Nx (where N is at (Nx, Ny))<BR>\n" );
document.write( "4.  Once Nx is found, plug in to (2) to find Ny.   You should find N to be at (Cx/3, Cy/3)<BR>\n" );
document.write( "6.  Write equations for |DN| and |DB| <BR>\n" );
document.write( "   |DN| = [  ((Cx+Bx)/4 - (Cx/3))^2 + (Cy/4 - Cy/3)^2 ] ^(1/2)<BR>\n" );
document.write( "        :<BR>\n" );
document.write( "      |DN| = [ ( 9Bx^2 - 6BxCx + Cx^2 - Cy^2 ) / 144 ] ^(1/2)<BR>\n" );
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document.write( "    |DB|  = [  ((Cx - 3Bx)/4 )^2 + (Cy/4)^2 ]^(1/2)<BR>\n" );
document.write( "        :<BR>\n" );
document.write( "      |DB| = [ (9Bx^2 - 6BxCx + Cx^2 + Cy^2) / 16 ] ^(1/2)\r<BR>\n" );
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document.write( "Writing |DN| / |DB|  and bringing everything under the radical sign, allows all the variables to cancel, leaving only:<BR>\n" );
document.write( "   ( 16/144 ) ^(1/2)<BR>\n" );
document.write( " = ( 4/12 )<BR>\n" );
document.write( " = (1 / 3 )\r<BR>\n" );
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document.write( "I started with the same method as @greenestamps.   I used a special case and worked it out numerically, and then I generalized it.   The proof must be done using a general case, not a special case.  <BR>\n" );
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document.write( "Also note that |DN| / |DB| = [AND] / [BMD]    where [T] denotes area of triangle T\r<BR>\n" );
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document.write( "which can be shown by making use of  A = (1/2)b*h (h = side*sin(alpha)), vertical angles are equal,  and these relationships that arise from properties of medians:<BR>\n" );
document.write( "[ACM] = [ABM] = (1/2)[ABC]<BR>\n" );
document.write( "[ABD] = [BMD] = (1/2)[ABM] = (1/4)[ABC]\r<BR>\n" );
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document.write( "—————————\r<BR>\n" );
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document.write( "6/16: Yes, tutor @ikleyn's solution is most appropriate (geometric).  I wish I had seen that construction.<BR>\n" );
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