ELMO SL 2018 G4

Let $ABCDEF$ be a hexagon inscribed in a circle $\Omega$ such that triangles $ACE$ and $BDF$ have the same orthocenter. Suppose that segments $BD$ and $DF$ intersect $CE$ at $X$ and $Y$ , respectively. Show that there is a point common to $\Omega$ , the circumcircle of $DXY$ , and the line through $A$ perpendicular to $CE$ .

Proposed by Michael Ren and Vincent Huang