OMCC 2000 Problem 5

Let $ABC$ be an acute-angled triangle. $C_{1}$ and $C_{2}$ are two circles of diameters $AB$ and $AC$ , respectively. $C_{2}$ and $AB$ intersect again at $F$ , and $C_{1}$ and $AC$ intersect again at $E$ . Also, $BE$ meets $C_{2}$ at $P$ and $CF$ meets $C_{1}$ at $Q$ . Prove that $AP=AQ$ .