CGMO 2011 Problem 2

The diagonals $AC,BD$ of the quadrilateral $ABCD$ intersect at $E$ . Let $M,N$ be the midpoints of $AB,CD$ respectively. Let the perpendicular bisectors of the segments $AB,CD$ meet at $F$ . Suppose that $EF$ meets $BC,AD$ at $P,Q$ respectively. If $MF\cdot CD=NF\cdot AB$ and $DQ\cdot BP=AQ\cdot CP$ , prove that $PQ\perp BC$ .

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