Question 64
Let $ABCD$ be a convex quadrilateral with $\angle DAB = \angle BCD = 90^{\circ}$ and $\angle ABC \gt \angle CDA$. Let $Q$ and $R$ be points on the segments $BC$ and $CD$, respectively, such that the line $QR$ intersects lines $AB$ and $AD$ at points $P$ and $S$, respectively. It is given that $PQ = RS$. Let the midpoint of $BD$ be $M$ and the midpoint of $QR$ be $N$. Prove that $M, N, A$ and $C$ lie on a circle.