SHARYGIN 2019 Problem 16

Let $AH_1$ and $BH_2$ be the altitudes of triangle $ABC$ . Let the tangent to the circumcircle of $ABC$ at $A$ meet $BC$ at point $S_1$ , and the tangent at $B$ meet $AC$ at point $S_2$ . Let $T_1$ and $T_2$ be the midpoints of $AS_1$ and $BS_2$ respectively. Prove that $T_1T_2$ , $AB$ and $H_1H_2$ concur.