KMO 2018 Problem 5

Let there be a convex quadrilateral $ABCD$ . The angle bisector of $\angle A$ meets the angle bisector of $\angle B$ , the angle bisector of $\angle D$ at $P, Q$ respectively. The angle bisector of $\angle C$ meets the angle bisector of $\angle D$ , the angle bisector of $\angle B$ at $R, S$ respectively. $P, Q, R, S$ are all distinct points. $PR$ and $QS$ meets perpendicularly at point $Z$ . Denote $l_A, l_B, l_C, l_D$ as the exterior angle bisectors of $\angle A, \angle B, \angle C, \angle D$ . Denote $E = l_A \cap l_B$ , $F= l_B \cap l_C$ , $G = l_C \cap l_D$ , and $H= l_D \cap l_A$ . Let $K, L, M, N$ be the midpoints of $FG, GH, HE, EF$ respectively.

Prove that the area of quadrilateral $KLMN$ is equal to $ZM \cdot ZK + ZL \cdot ZN$ .