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incircles, semicircles, and lines concurrent

Source: 2014 Sharygin Geometry Olympiad Final Round 10.6

August 3, 2018
geometryconcurrencyconcurrent

Problem Statement

The incircle of a non-isosceles triangle ABCABC touches ABAB at point C′C'. The circle with diameter BC′BC' meets the incircle and the bisector of angle BB again at points A1A_1 and A2A_2 respectively. The circle with diameter AC′AC' meets the incircle and the bisector of angle AA again at points B1B_1 and B2B_2 respectively. Prove that lines AB,A1B1,A2B2AB, A_1B_1, A_2B_2 concur.
(E. H. Garsia)
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