KöMaL 2015 B4708

$O$ is the centre of the circumscribed circle of triangle $ABC$ , and $M$ is the orthocentre. Point $A$ is reflected in the perpendicular bisector of side $BC$ , $B$ is reflected in the perpendicular bisector of side $CA$ , and finally $C$ is reflected in the perpendicular bisector of side $AB$ . The reflections are denoted by $A_1$, $B_1$, $C_1$, respectively. Let $K$ be the centre of the inscribed circle of triangle $A_1B_1C_1$. Prove that point $O$ bisects line segment $MK$ .