Brazilian Math Olympiad

Let $ABC$ be an acute triangle and $H$ is orthocenter. Let $D$ be the intersection of $BH$ and $AC$ and $E$ be the intersection of $CH$ and $AB$. The circumcircle of $ADE$ meets the circumcircle of $ABC$ at $F\ne A$. Prove that the angle bisectors of $\angle BFC$ and $\angle BHC$ concur at a point on line $BC$.