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Geometry Mathley 10.2 concurrent, collinear radical centers

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June 7, 2020
geometrycollinearconcurrentcircumcircleincircleradical axis

Problem Statement

Let ABCABC be an acute triangle, not isoceles triangle and (O),(I)(O), (I) be its circumcircle and incircle respectively. Let A1A_1 be the the intersection of the radical axis of (O),(I)(O), (I) and the line BCBC. Let A2A_2 be the point of tangency (not on BCBC) of the tangent from A1A_1 to (I)(I). Points B1,B2,C1,C2B_1,B_2,C_1,C_2 are defined in the same manner. Prove that (a) the lines AA2,BB2,CC2AA_2,BB_2,CC_2 are concurrent. (b) the radical centers circles through triangles BCA2,CAB2BCA_2, CAB_2 and ABC2ABC_2 are all on the line OIOI.
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