Stuck at a similar triangles problem.

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BoykoLenkov

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Apr 6, 2019
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Hello. It would be great if you helped with the following problem:

In the triangle ABC, M is the middle of AB. AN is the bisector through A and AB:AC = 2:3. If O is the intersection point of AN and CM, find CO:OM and AO:ON.

I've got a vague idea of how to solve the problem. I'm supposed to use similar triangles and the angle bisector theorem.
 
What does the angle bisector theorem tell you about this triangle?

What extra line might you draw that would result in similar triangles?

There are probably many ways to solve the problem, so I'd like to start with your ideas, not mine. Please show whatever you're able to do, and we'll work from there.
 
The angle bisector theorem tells me that CN:NB=AC:AB=3:2. CM is also a median, which means it splits AB in half. The angles CAL and LAB are equal because of the bisector. If I drew a line through M, parallel to BC, it would give me a midline.

Ps: Sorry for taking so long to respond.
 
You haven't mentioned any similar triangles, so let's take what you did mention, the midline through M. What similar triangles does that form? Can you use any of those to get in the direction you need to go?
 
If we name the point, where the midline intersects AC Q, then the triangles AQM and ABC are similar. From then we see that AM:AB=QM:BC=1:2
 
That's true; but does it help you solve the problem? They're asking for some ratios involving point O; can you see any similar triangles using that point? Or can you take the fact you've found and use it to go further?
 
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