BALKAN-SL 2015 G2

Let $ABC$ be a triangle with circumcircle $\omega$ . Point $D$ lies on the arc $BC$ of $\omega$ and is different than $B,C$ and the midpoint of arc $BC$ . Tangent of $\Gamma$ at $D$ intersects lines $BC$ , $CA$ , $AB$ at $A',B',C'$ , respectively. Lines $BB'$ and $CC'$ intersect at $E$ . Line $AA'$ intersects the circle $\omega$ again at $F$ . Prove that points $D,E,F$ are collinear.

(Saudi Arabia)