OMM 1996 Problem 1

Let $P$ and $Q$ be the points on the diagonal $BD$ of a quadrilateral $ABCD$ such that $BP = PQ = QD$ . Let $AP$ and $BC$ meet at $E$ , and let $AQ$ meet $DC$ at $F$ . (a) Prove that if $ABCD$ is a parallelogram, then $E$ and $F$ are the midpoints of the corresponding sides. (b) Prove the converse of (a).