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3 equal circles or radius = inradius r, rioplatense line concurrency 2015 p6

Source: Rioplatense Olympiad 2015 level 3 P6

September 3, 2018
geometrycirclesinradiusconcurrencyconcurrent

Problem Statement

Let ABCA B C be an acut-angles triangle of incenter II, circumcenter OO and inradius r.r. Let ω\omega be the inscribed circle of the triangle ABCA B C. A1A_1 is the point of ω\omega such that AIA1OA IA_1O is a convex trapezoid of bases AOA O and IA1IA_1. Let ω1\omega_1 be the circle of radius rr which goes through A1A_1, tangent to the line ABA B and is different from ω\omega . Let ω2\omega_2 be the circle of radius rr which goes through A1A_1, is tangent to the line ACA C and is different from ω\omega . Circumferences ω1\omega_1 and ω2\omega_2 they are cut at points A1A_1 and A2A_2. Similarly are defined points B2B_2 and C2C_2. Prove that the lines AA2,BB2A A_2, B B_2 and CC2CC2 they are concurrent.
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