ELMO SL 2014 G10

We are given triangles $ABC$ and $DEF$ such that $D\in BC, E\in CA, F\in AB$ , $AD\perp EF, BE\perp FD, CF\perp DE$ . Let the circumcenter of $DEF$ be $O$ , and let the circumcircle of $DEF$ intersect $BC,CA,AB$ again at $R,S,T$ respectively. Prove that the perpendiculars to $BC,CA,AB$ through $D,E,F$ respectively intersect at a point $X$ , and the lines $AR,BS,CT$ intersect at a point $Y$ , such that $O,X,Y$ are collinear.

Proposed by Sammy Luo