IMO 1982 Problem 2

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$ , $a_{2}$ , $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$ . Let $M_{i}$ be the midpoint of the side $a_{i}$ , and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$ . Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$ . Prove that the lines $M_{1}S_{1}$ , $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.