Školjka

%V0 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.