Given a triangle $ABC$ in which the angle $B$ is equal to $60^\circ$. A circle inscribed in a triangle with a center $I$ touches the side $AC$ at point $K$. A line passing through the points of touching of this circle with the other sides of the triangle intersects the its circumcircle at points $M$ and $N$. Prove that the ray $KI$ divides the arc $MN$ in half.