TH-TMO 2013 Problem 9

Let $ABCD$ be a convex quadrilateral, and let $M$ and $N$ be midpoints of sides $AB$ and $CD$ respectively. Point $P$ is chosen on $CD$ so that $MP \perp CD$ , and point $Q$ is chosen on $AB$ so that $NQ \perp AB$ . Show that $AD \parallel BC$ if and only if $\frac{AB}{CD} =\frac{MP}{NQ}$ .