Question 177
Let $ABC$ be an acute scalene triangle with $AC \gt BC \gt AB$. Let the orthocentre be $H$ and circumcentre be $O$. Suppose that lines $BO$ and $CH$ intersect at a point $D$. Point $E$ (where $E \neq C$) lies on side $AC$ so that $OECD$ is cyclic. Point $F$ (where $F \neq C$) lies on side $BC$ such that $CE = FE$. Prove that $B H D F$ is cyclic.
(The orthocentre of a triangle is the point of intersection of its altitudes.)
(The orthocentre of a triangle is the point of intersection of its altitudes.)