BALKAN-SL 2013 G2

Let $ABCD$ be a quadrilateral, let $O$ be the intersection point of diagonals $AC$ and $BD$ , and let $P$ be the intersection point of sides $AB$ and $CD$ . Consider the parallelograms $AODE$ and $BOCF$ . Prove that $E, F$ and $P$ are collinear.