Question 156
Let $ABC$ be an acute scalene triangle with orthocenter $H$. Let $M$ be the midpoint of $BC$, and suppose that the line through $H$ perpendicular to $AM$ intersects $AB$ and $AC$ at points $E$ and $F$ respectively. Denote by $O$ the circumcenter of triangle $AEF$, and $D$ the foot of the perpendicular from $H$ to $AM$. Prove that the line $AO$ intersects the perpendicular from $D$ to $BC$ at a point on the circumcircle of triangle $ABC$.