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Radical Center lies on OI

Source: Sharygin geometry olympiad 2016, grade 10, Final Round, Problem 8.

August 5, 2016
geometrygeometry proposed

Problem Statement

Let ABCABC be a non-isosceles triangle, let AA1AA_1 be its angle bisector and A2A_2 be the touching point of the incircle with side BCBC. The points B1,B2,C1,C2B_1,B_2,C_1,C_2 are defined similarly. Let OO and II be the circumcenter and the incenter of triangle ABCABC. Prove that the radical center of the circumcircle of the triangles AA1A2,BB1B2,CC1C2AA_1A_2, BB_1B_2, CC_1C_2 lies on the line OIOI.
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