KöMaL 2007 B3988

The midpoints of the sides of a convex pentagon are $F_1$, $F_2$, $F_3$, $F_4$, $F_5$, in this order. (The vertex $A$ is between $F_5$ and $F_1$.) Let $P$ be the point in the plane for which the quadrilateral $PF_2 F_3 F_4$ is a parallelogram. Prove that the quadrilateral $PF_5 AF_1$ is also a parallelogram.