Question 174
Let $ABCD$ be a quadrilateral. Let point $P$ be on side $BC$, and point $Q$ be on side $CD$ such that $2PB = AB$ and $2QD = AD$. Let $M$ be the midpoint of segment $BD$, and $N$ be the midpoint of segment $PQ$. If $4MN = BD$, prove that $ABCD$ is a cyclic quadrilateral.