Question 93
Let $ABCD$ be a convex quadrilateral, and let $P, Q, R, S$ be points on the sides $AB, BC, CD$, and $DA$, respectively. Let the line segments $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS, BQOP, CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC, PQ$, and $RS$ are either concurrent or parallel to each other.