Question 185
In an acute-angled triangle $ABC$, let us denote $H$ its orthocenter and $I$ its incenter. Let $D$ be the perpendicular projection of $I$ on the line $BC$, and $E$ be the image of point $A$ in symmetry with center $I$. Furthermore, $F$ is the perpendicular projection of the point $H$ on the line $ED$. Prove that the points $B$, $H$, $F$ and $C$ lie on one circle. (Patrik Bak)