BALKAN-SL 2018 G1

In an acute triangle $A B C$, the midpoint of the side $B C$ is $M$ and the centers of the excircles relative to $M$ of the triangles $A M B$ and $A M C$ are $D$ and $E$ respectively. The circumcircle of the triangle $A B D$ meets line $B C$ at $B$ and $F$. The circumcircle of the triangle $A C E$ meets line $B C$ at $C$ and $G$. Prove that $B F=C G$. (Romania)