ISL 1995 G8

Suppose that $ABCD$ is a cyclic quadrilateral. Let $E = AC\cap BD$ and $F = AB\cap CD$ . Denote by $H_{1}$ and $H_{2}$ the orthocenters of triangles $EAD$ and $EBC$ , respectively. Prove that the points $F$ , $H_{1}$ , $H_{2}$ are collinear.

Original formulation:

Let $ABC$ be a triangle. A circle passing through $B$ and $C$ intersects the sides $AB$ and $AC$ again at $C'$ and $B',$ respectively. Prove that $BB'$ , $CC'$ and $HH'$ are concurrent, where $H$ and $H'$ are the orthocentres of triangles $ABC$ and $AB'C'$ respectively.