PAMO 2013 Problem 3

Let $ABCDEF$ be a convex hexagon with $\angle A= \angle D$ and $\angle B=\angle E$ . Let $K$ and $L$ be the midpoints of the sides $AB$ and $DE$ respectively. Prove that the sum of the areas of triangles $FAK$ , $KCB$ and $CFL$ is equal to half of the area of the hexagon if and only if $$\frac{BC}{CD}=\frac{EF}{FA}.$$