BALKAN 2012 Problem 1

$Let$A, B$and$C$be points lying on a circle$\Gamma$with centre$O$. Assume that$\angle A B C>90$. Let$D$be the point of intersection of the line$A B$with the line perpendicular to$A C$at$C$. Let$l$be the line through$D$which is perpendicular to$A O$. Let$E$be the point of intersection of$l$with the line$A C$, and let$F$be the point of intersection of$\Gamma$with$l$that lies between$D$and$E$. Prove that the circumcircles of triangles$B F E$and$C F D$are tangent at$F$.