There are similar because each side of the black triangle is bisected by the red triangle evenly.
Here's a better picture that I've attached. I've discovered that when the bisectors of two sides are joined, the line joining them is parallel to the third side. This makes it easy to form parallelograms, and prove that the sides of the inner triangle are half the length of the sides they're parallel to. Now I just need a proof that shows why the lines are parallel!!