Q:

Find how a point divides the sides of a triangle

Given a triangle $AB$C. On the sides $BC$, $CA$ and $AB$ points $A_1$, $B_1$ and $C_1$ have been selected respectively. Medians $A_1A_2$, $B_1B_2$ and $C_1C_2$ of the triangle $A_1B_1C_1$ are parallel to $AB$, $BC$ and $CA$ respectively. How can I find in what relationship do points $A_1$, $B_1$ and $C_1$ divide the sides of the triangle $ABC$?

A:

Since affine maps preserve midpoints and parallel lines we may assume without loss of generality that $ABC$ is equilateral. The following configuration trivially fulfills the wanted constraints

hence $A_1,B_1,C_1$ all lie at two thirds of the corresponding side.

