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Incenter lies on common tangent

Source: All-Russian MO 1999

December 31, 2012
geometryincentergeometry unsolved

Problem Statement

A triangle ABCABC is inscribed in a circle SS. Let A0A_0 and C0C_0 be the midpoints of the arcs BCBC and ABAB on SS, not containing the opposite vertex, respectively. The circle S1S_1 centered at A0A_0 is tangent to BCBC, and the circle S2S_2 centered at C0C_0 is tangent to ABAB. Prove that the incenter II of △ABC\triangle ABC lies on a common tangent to S1S_1 and S2S_2.
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