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O_1S_1, O_2S_2 and O_3S_3 are concurrent

Source: Baltic Way 1994

December 22, 2011
geometryincentergeometry proposed

Problem Statement

The inscribed circle of the triangle A1A2A3A_1A_2A_3 touches the sides A2A3,A3A1,A1A2A_2A_3,A_3A_1,A_1A_2 at points S1,S2,S3S_1,S_2,S_3, respectively. Let O1,O2,O3O_1,O_2,O_3 be the centres of the inscribed circles of triangles A1S2S3,A2S3S1,A3S1S2A_1S_2S_3, A_2S_3S_1,A_3S_1S_2, respectively. Prove that the straight lines O1S1,O2S2,O3S3O_1S_1,O_2S_2,O_3S_3 intersect at one point.
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