MEMO 2016 Problem 5

Let $ABC$ be an acute triangle for which $AB \neq AC$ , and let $O$ be its circumcenter. Line $AO$ meets the circumcircle of $ABC$ again in $D$ , and the line $BC$ in $E$ . The circumcircle of $CDE$ meets the line $CA$ again in $P$ . The lines $PE$ and $AB$ intersect in $Q$ . Line passing through $O$ parallel to the line $PE$ intersects the $A$ -altitude of $ABC$ in $F$ .

Prove that $FP = FQ$ .