Question 8
Let $H$ be the point of intersection of the altitudes $AP$ and $CQ$ of an acute-angled triangle $ABC$. On the median $BM$ points $E$ and $F$ are chosen so that $\angle APE = \angle BAC$, $\angle CQF = \angle BCA$, where the point $E$ lies inside the triangle $APB$, and the point $F$ lies inside the triangle $CQB$. Prove that the lines $AE$, $CF$ and $BH$ are concurrent.