Middle European Mathematical Olympiad 2016 Problem T-5

Let $ABC$ be an acute-angled triangle with $AB \neq AC$, and let $O$ be its circumcentre. The line $AO$ intersects the circumcircle $\omega$ of $ABC$ a second time in point $D$, and the line $BC$ in point $E$. The circumcircle of $CDE$ intersects the line $CA$ a second time in point $P$. The line $PE$ intersects the line $AB$ in point $Q$. The line through $O$ parallel to $PE$ intersects the altitude of the triangle $ABC$ that passes through $A$ in point $F$.

Prove that $FP = FQ$.