ELMO SL 2013 G2

Let $ABC$ be a scalene triangle with circumcircle $\Gamma$ , and let $D$ , $E$ , $F$ be the points where its incircle meets $BC$ , $AC$ , $AB$ respectively. Let the circumcircles of $\triangle AEF$ , $\triangle BFD$ , and $\triangle CDE$ meet $\Gamma$ a second time at $X,Y,Z$ respectively. Prove that the perpendiculars from $A,B,C$ to $AX,BY,CZ$ respectively are concurrent.

Proposed by Michael Kural