MathNet 2017-part-04 — 176 problems (EN–VI) Question 172

Let $ABCD$ be any convex quadrangle and let $P, Q, R, S$ be points on the segments $AB, BC, CD$, and $DA$, respectively. It is given that the segments $PR$ and $QS$ dissect $ABCD$ into four quadrangles, each of which has perpendicular diagonals. Show that the points $P, Q, R, S$ are concyclic.

Nikolai Beluhov, Bulgaria