KöMaL 2010 B4280

$M$ is the midpoint of the arc $AB$ containing vertex $C$ on the circumscribed circle $k$ of triangle ABC . $J$ is the centre of the escribed circle drawn to side $AB$. The perpendicular drawn to angle bisector $CJ$ at point $J$ intersects line $AC$ at $D$ and line $BC$ at $E$. Line $MJ$ intersects circle $k$ again at $F$. Prove that the circle passing through points $D$, $E$, $F$ touches the lines $AC$ and $BC$, and also touches the circle $k$.