MathNet 2014 Question 8

Let $H$ be an intersection point of the altitudes $AA_1$, $BB_1$, $CC_1$ of an acute-angled triangle $ABC$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively.
Prove that $MN$ is the perpendicular bisector of the segment $B_1C_1$.