document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=885619>Question 885619</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have a problem as follows :\r<BR>\n" );
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document.write( "Point M and N are taken on sides AB & AC of the triangle ABC such that AN = 2NC<BR>\n" );
document.write( "and AM = 2MB. If CM and BN intersect in O then Prove that 5CO = 3CM </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #535411 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=535411\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here is the drawing showing my interpretation of your problem description.<BR>\n" );
document.write( "I added some green lines parallel to triangle base BC, cutting sides AB and AC into 3 congruent segments.<BR>\n" );
document.write( "I also added those little red marks indicating that the 3 segments of each side are congruent.<BR>\n" );
document.write( "The green lines are parallel to each other and to BC, of course,<BR>\n" );
document.write( "since the segments they cut in AB and AC are proportional,<BR>\n" );
document.write( "so I added the arrows indicating that.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-1%2C10%2C-1%2C10%2C%0D%0Aline%280%2C0%2C3%2C9%29%2Cline%280%2C0%2C9%2C0%29%2Cline%283%2C9%2C9%2C0%29%2C%0D%0Aline%281%2C3%2C9%2C0%29%2Cline%287%2C3%2C0%2C0%29%2C%0D%0Agreen%28line%281%2C3%2C7%2C3%29%29%2Cgreen%28line%282%2C6%2C5%2C6%29%29%2C%0D%0Agreen%28arrow%282%2C3%2C4.5%2C3%29%29%2Cgreen%28arrow%282%2C6%2C4.5%2C6%29%29%2C%0D%0Aarrow%282%2C0%2C4.5%2C0%29%2C%0D%0Ared%28line%280.3%2C1.5%2C0.7%2C1.5%29%29%2Cred%28line%281.3%2C4.5%2C1.7%2C4.5%29%29%2C%0D%0Ared%28line%282.3%2C7.5%2C2.7%2C7.5%29%29%2C%0D%0Ared%28line%287.8%2C1.4%2C8.2%2C1.4%29%29%2Cred%28line%287.7%2C1.6%2C8.1%2C1.6%29%29%2C%0D%0Ared%28line%285.8%2C4.4%2C6.2%2C4.4%29%29%2Cred%28line%285.7%2C4.6%2C6.1%2C4.6%29%29%2C%0D%0Ared%28line%283.8%2C7.4%2C4.2%2C7.4%29%29%2Cred%28line%283.7%2C7.6%2C4.1%2C7.6%29%29%2C%0D%0Alocate%283%2C9.4%2CA%29%2Clocate%289%2C0%2CB%29%2Clocate%28-0.3%2C0%2CC%29%2C%0D%0Alocate%284.1%2C1.8%2CO%29%2Clocate%280.7%2C3.25%2CN%29%2Clocate%287.1%2C3.25%2CM%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Triangles ABC and AMN are similar because they share an angle at A,<BR>\n" );
document.write( "and the corresponding sides that end in A are proportional:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AM%2FAB=AM%2F%28AM%2BMB%29=AM%2F%28AM%2B2AM%29=AM%2F3AM=2%2F3 ALIGN=MIDDLE  ALT=\"AM%2FAB=AM%2F%28AM%2BMB%29=AM%2F%28AM%2B2AM%29=AM%2F3AM=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and similarly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AN%2FAC=2%2F3 ALIGN=MIDDLE  ALT=\"AN%2FAC=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, AMN is a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F4 ALIGN=MIDDLE  ALT=\"3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > scale version of ABC, and so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MN%2FBC=2%2F3 ALIGN=MIDDLE  ALT=\"MN%2FBC=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > too.<BR>\n" );
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document.write( "Triangles BCO and NMO are also similar,<BR>\n" );
document.write( "because they have 3 pairs of congruent angles.<BR>\n" );
document.write( "Angle BCM (the same as BCO) and angle MNB (the same as MNO) are congruent,<BR>\n" );
document.write( "because they are alternate interior angles formed when parallel lines BC and MN are intersected by transversal line CM.<BR>\n" );
document.write( "Similarly, angle CBN (same as CBO) and angle MNB (same as MNO) are congruent.<BR>\n" );
document.write( "Of course, the angles at O (NOM and BOC) also congruent, because they are vertical angles, formed as lines CM and BN intersect.<BR>\n" );
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document.write( "Note: We only need to prove congruency of 2 out of the 3 pairs of corresponding angles, so you do not have to give reasons for congruency of all 3 pairs, but it was easy enough.<BR>\n" );
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document.write( "Since triangles BCO and NMO are similar, their corresponding sides are proportional.<BR>\n" );
document.write( "Since we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MN%2FBC=2%2F3 ALIGN=MIDDLE  ALT=\"MN%2FBC=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the same ratio applies to the other two pairs of corresponding sides.<BR>\n" );
document.write( "In particular, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CO%2FMO=2%2F3 ALIGN=MIDDLE  ALT=\"CO%2FMO=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=MO=%282%2F3%29%2ACO ALIGN=MIDDLE  ALT=\"MO=%282%2F3%29%2ACO\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CM=CO%2BMO=CO%2B%282%2F3%29%2ACO=%281%2B2%2F3%29%2ACO=%285%2F3%29%2ACO ALIGN=MIDDLE  ALT=\"CM=CO%2BMO=CO%2B%282%2F3%29%2ACO=%281%2B2%2F3%29%2ACO=%285%2F3%29%2ACO\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CM=%285%2F3%29%2ACO ALIGN=MIDDLE  ALT=\"CM=%285%2F3%29%2ACO\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3CM=5CO ALIGN=MIDDLE  ALT=\"3CM=5CO\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
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