IMO Shortlist 1982 problem 13


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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.
Source: Međunarodna matematička olimpijada, shortlist 1982