SHARYGIN 2016 Problem 22

Let $M_A, M_B, M_C$ be the midpoints of the sides $BC, CA, AB$ respectively of a non-isosceles triangle $ABC$ . Points $H_A, H_B, H_C$ lie on the corresponding sides, different from $M_A, M_B, M_C$ such that $M_AH_B=M_AH_C,$ $M_BH_A=M_BH_C,$ and $M_CH_A=M_CH_B$ . Prove that $H_A, H_B, H_C$ are the feet of the corresponding altitudes.