CL-MO 2021 Problem 4

Consider quadrilateral $ABCD$ with $|DC| > |AD|$ . Let $P$ be a point on $DC$ such that $PC = AD$ and let $Q$ be the midpoint of $DP$ . Let $L_1$ be the line perpendicular on $DC$ passing through $Q$ and let $L_2$ be the bisector of the angle $\angle ABC$ . Let us call $X = L_1 \cap L_2$ . Show that if quadrilateral is cyclic then $X$ lies on the circumcircle of $ABCD.$