BALKAN 2006 Problem 2

Let $ABC$ be a triangle and $m$ a line which intersects the sides $AB$ and $AC$ at interior points $D$ and $F$ , respectively, and intersects the line $BC$ at a point $E$ such that $C$ lies between $B$ and $E$ . The parallel lines from the points $A$ , $B$ , $C$ to the line $m$ intersect the circumcircle of triangle $ABC$ at the points $A_1$ , $B_1$ and $C_1$ , respectively (apart from $A$ , $B$ , $C$ ). Prove that the lines $A_1E$ , $B_1F$ and $C_1D$ pass through the same point.

Greece