Why is $\Delta ABC\sim \Delta AEC\sim \Delta FBC$? I see that they have two angles in common and also share a side. But this doesn't make them similar as far as I am concerned.
Rectangles of the same color are equal in area, angles of the same color are equal in size.
Consider triangles $ABC$ and $ACE$. They have $\angle A$ in common and $\angle ACB=\angle AEC$ by construction. Hence they are similar.
An analogous reasoning can be made to show that $ABC\sim FCB$.