2016 European Girls' Mathematical Olympiad

Let $ABCD$ be a cyclic quadrilateral, and let diagonals $AC$ and $BD$ intersect at $X$. Let $C_1$, $D_1$ and $M$ be the midpoints of segments $CX$, $DX$ and $CD$, respectively. Lines $A{D}_1$ and $B{C}_1$ intersect at $Y$, and line $MY$ intersects diagonals $AC$ and $BD$ at different points $E$ and $F$, respectively. Prove that line $XY$ is tangent to the circle through $E$, $F$ and $X$.