OMCC 2015 Problem 5

Let $ABC$ be a triangle such that $AC=2AB$ . Let $D$ be the point of intersection of the angle bisector of the angle $CAB$ with $BC$ . Let $F$ be the point of intersection of the line parallel to $AB$ passing through $C$ with the perpendicular line to $AD$ passing through $A$ . Prove that $FD$ passes through the midpoint of $AC$ .