Let $P_{\infty}$ be the point of infinity on the lines $BD$ and $CE$. Let $l$ be the angle bisector of $\angle BAC$, clearly $l,BD,CE$ share the same point at infinity becuase $l \perp DE$. Let $BC$ meet $DE$ at $H$, $BC$ meets $l$ at $G$ and $AF$ meets $BC$ at $J$. $-1=(H, B; G, C) \overset{P_{\infty}}{=} (H, D; A, E) \overset{F}{=} (H, C; J, B)=(H, B; J, C) \implies G=J$ Thus $F$ lies on $l$ and we are done becuase its known that $l \perp DE$.