The converse of the triangle proportionality theorem states that if a line divides two sides of a triangle proportionally, then it is parallel to the

___ third side. This can be nicely proven with vector geometry. I would like to draw you a diagram but I can't work out how to upload an image, so I'll have to describe it to you and you can draw it.

Imagine a point with two vectors protruding from it, a and b (these letters should all have bars on them really to show their vector nature). They have an acute angle between them.

The ends of these two vectors are joined up to produce s third vector, going from the end of a to the end of b. We find that this vector c = b - a.

Next, draw a fourth vector from the midpoint of a to the midpoint of b. We find this vector d = 0.5b - 0.5a = 0.5(b - a).

This shows that c = 2d. As c is a multiple of d, c and d must be parallel. Hence the statement is true.

I hope this helps you :)