KöMaL 2014 B4624

In a trapezium $ABCD$ , let $E$ and $F$ denote the midpoints of the bases $AB$ and $CD$ , respectively, and let $O$ denote the intersection of the diagonals. A line parallel to the bases intersects the line segments $OA$ , $OE$ and $OB$ at points $M$ , $N$ and $P$ , respectively. Show that the quadrilaterals $APCN$ and $BNDM$ have equal areas.