Question 19
Let $ABC$ be an acute triangle and $H$ be the perpendicular foot at the side $BC$ from $A$. Let $D$ and $E$ be points on the segments $AB$ and $AC$, $F$ and $G$ be the perpendicular feet at $BC$ from $D$ and $E$ respectively. Let $P$ be the perpendicular foot at the segment $DH$ from $E$. Suppose that $DG, EF, AH$ are concurrent. Show that $\angle APE = \angle CPE$.