Question 84
Let $ABCD$ be a convex quadrilateral. Let $M$ and $N$ be the midpoints of $AB$ and $CD$, respectively. Let $P$ be the point on the line $CD$ such that $MP$ is perpendicular to $CD$. Let $Q$ be the point on the line $AB$ such that $NQ$ is perpendicular to $AB$. Prove that $AD$ is parallel to $BC$ if and only if $\frac{AB}{CD} = \frac{MP}{NQ}$.