Question 885619: I have a problem as follows :
Point M and N are taken on sides AB & AC of the triangle ABC such that AN = 2NC
and AM = 2MB. If CM and BN intersect in O then Prove that 5CO = 3CM
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! Here is the drawing showing my interpretation of your problem description.
I added some green lines parallel to triangle base BC, cutting sides AB and AC into 3 congruent segments.
I also added those little red marks indicating that the 3 segments of each side are congruent.
The green lines are parallel to each other and to BC, of course,
since the segments they cut in AB and AC are proportional,
so I added the arrows indicating that.
Triangles ABC and AMN are similar because they share an angle at A,
and the corresponding sides that end in A are proportional:
 , and similarly  .
So, AMN is a  scale version of ABC, and so  too.
Triangles BCO and NMO are also similar,
because they have 3 pairs of congruent angles.
Angle BCM (the same as BCO) and angle MNB (the same as MNO) are congruent,
because they are alternate interior angles formed when parallel lines BC and MN are intersected by transversal line CM.
Similarly, angle CBN (same as CBO) and angle MNB (same as MNO) are congruent.
Of course, the angles at O (NOM and BOC) also congruent, because they are vertical angles, formed as lines CM and BN intersect.
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.
Since triangles BCO and NMO are similar, their corresponding sides are proportional.
Since we know that , the same ratio applies to the other two pairs of corresponding sides.
In particular,  <---> .
So,  , and
 <---> .
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