International Mathematical Olympiad 1982 Problem 2

A non-isosceles triangle $A_1A_2A_3$ is given with sides $a_1, a_2, a_3$ ($a_i$ is the side opposite $A_i$). For all $i = 1, 2, 3, M_i$ is the midpoint of side $a_i$, and $T_i$ is the point where the incircle touches side $a_i$. Denote by $S_i$ the reflection of $T_i$ in the interior bisector of angle $A_i$. Prove that the lines $M_1S_1$, $M_2S_2$, and $M_3S_3$ are concurrent.