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prove that 3 circles have 2 common points

Source: 2015 Sharygin Geometry Olympiad Correspondence Round P14

August 2, 2018
geometrycirclessymmetry

Problem Statement

Let ABCABC be an acute-angled, nonisosceles triangle. Point A1,A2A_1, A_2 are symmetric to the feet of the internal and the external bisectors of angle AA wrt the midpoint of BCBC. Segment A1A2A_1A_2 is a diameter of a circle α\alpha. Circles β\beta and γ\gamma are defined similarly. Prove that these three circles have two common points.
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