MathNet 2009-part-04 — 175 problems (EN–VI) Question 70

In an acute triangle $ABC$ let $BE$ and $CF$ be altitudes. Two circles through $A$ and $F$ touch the line $BC$ at $P$ and $Q$, such that $B$ lies between $C$ and $Q$. Prove that the lines $PE$ and $QF$ meet on the circumcircle of $AEF$.