Question 9
Let $ABCD$ be a quadrilateral inscribed in a circle ($O$) and some point $I$ lies inside $ABCD$. Consider the lines $d_1, d_2, d_3, d_4$ pass through the midpoints of segments $IA, IB, IC, ID$ respectively and perpendicular to lines $OA, OB, OC, OD$. Line $d_1$ cuts $d_2$ at $P$, line $d_2$ cuts $d_3$ at $Q$, line $d_3$ cuts $d_4$ at $R$ and line $d_4$ cuts $d_1$ at $S$. Suppose that the quadrilateral $PQRS$ is convex and points $I, O$ lie inside it. Prove that $PQRS$ circumscribes a circle.