TUYMAADA 2016 Problem 3

Altitudes $AA_1$ , $BB_1$ , $CC_1$ of an acute triangle $ABC$ meet at $H$ . $A_0$ , $B_0$ , $C_0$ are the midpoints of $BC$ , $CA$ , $AB$ respectively. Points $A_2$ , $B_2$ , $C_2$ on the segments $AH$ , $BH$ , $HC_1$ respectively are such that $\angle A_0B_2A_2 = \angle B_0C_2B_2 = \angle C_0A_2C_2 =90^\circ$ . Prove that the lines $AC_2$ , $BA_2$ , $CB_2$ are concurrent.