MathNet 2026 — Part 08 Question 3

Let $ABCD$ be a convex quadrilateral with $AC \perp BD$. Prove that there exist points $P, Q, R, S$ on the sides $AB, BC, CD, DA$, respectively, such that $PR \perp QS$ and the area of quadrilateral $PQRS$ is exactly half that of $ABCD$.