SHARYGIN 2015 Problem 11

Let $H$ be the orthocenter of an acute-angled triangle A $BC$ . The perpendicular bisector to segment $BH$ meets $BA$ and $BC$ at points $A_0, C_0$ respectively. Prove that the perimeter of triangle $A_0OC_0$ ( $O$ is the circumcenter of triangle $ABC$ ) is equal to $AC$ .