MathNet 2014 Question 26

Let $ABC$ be a triangle with $\angle A \lt \angle B \leq \angle C$, $M$ and $N$ the midpoints of sides $CA$ and $AB$, respectively, and $P$ and $Q$ the projections of $B$ and $C$ on the medians $CN$ and $BM$, respectively. Prove that the quadrilateral $MNPQ$ is cyclic.

Supplementary illustration.