Jtdylktuy

I had a math exam today about geometry and similar triangles.
One of our math puzzle wanted us to proves something. Now I’ll explain that for you and if you help me I won’t lose 2 points of my midterm exam! So imagine that I’m your student and you’ve asked this question and I’ve answered like that.

QUESTION: We have two similar triangles. Prove that ratio of correspondent medians is same as ratio of correspondent sides.

MY ANSWER: We suppose two similar triangles, $triangle ABC$ and $triangle A’B’C’$. and also I did not mention that sides are equal! I mean $AB neq A’B’$ , $AC neq A’C’$ , $BC neq B’C’$.
And then I draw diagram 2. You can take a look here.

Actually I combined shapes in diagram 1, and I just draw diagram 2 in my exam paper. (I changed name of points in diagrams to explain what I answered better)

I wrote that we know:
$$triangle ABCthicksim triangle AMN$$
$$MN parallel BC$$
$$BH=HC$$
$$MO=ON$$
$AO space, AH$ are medians

So I continued based on thales theorem:
$$frac{AM}{MB}=frac{AO}{OH}$$
$$frac{AN}{NC}=frac{AO}{OH}$$
Thus $$frac{AM}{MB}=frac{AN}{NC}$$
On the other side :
$$frac{AM}{MB}=frac{AN}{NC}=frac{AO}{OH}$$
And finally he gave me a big beautiful zero! I don’t know why and I hadn’t a change to talk to him. What’s your Idea? Is my answer OK? If yes tell me why. Because I’m going to convince him.

Staying with diagram 1, by similarity$$frac{AB}{BC}=frac{A'B'}{B'C'}$$But since $M$ and $M'$ are midpoints we know$$frac{AB}{BM}=frac{A'B'}{B'M'}$$And we're given $$angle B=angle B'$$Therefore (Euclid, Elements VI, 4 & 6)$$triangle ABMsimtriangle A'B'M'$$from which it follows that$$frac{AB}{A'B'}=frac{AM}{A'M'}$$

I'm not sure it helped to superimpose the triangles (diagram 2). Your concluding proportion has lines $MB$, $NC$, $OH$, which are differences of corresponding sides, but doesn't the question call for $AB$, $AC$, $AH$ instead? You're almost there, and the teacher's zero seems a bit harsh, but you may not win your case.

I think you have a reasonable idea here, but your proof is incomplete. In order to apply Thales' theorem in this way, you need to know that $A$, $O$, and $H$ are collinear, and you haven't given any reason why they should be.

Notice that you haven't ever used the fact that $O$ and $H$ are midpoints. This is what you will need to prove collinearity.