Question 191
Let $I$, $O$, $H$ be the incenter, circumcenter, orthocenter of an acute triangle $ABC$, respectively, and the inscribed circle of $\triangle ABC$ touch side $BC$ at $D$. Suppose that $\angle B \lt \angle C$, and $AO$ is parallel to $HD$. Show that the four points $E$, $F$, $I$, $O$ lie on a common circle, where $E$ is the intersecting point of lines $OD$ and $AH$, and $F$ is the midpoint of line segment $CI$.