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A_1B_1//<B bisector, circle with center I (2021 Novosibirsk Oral Geo Oly 9.7)

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May 25, 2021
parallelincentergeometry

Problem Statement

A circle concentric with the inscribed circle of ABCABC intersects the sides of the triangle at six points forming a convex hexagon A1A2B1B2C1C2A_1A_2B_1B_2C_1C_2 (points C1C_1 and C2C_2 on the ABAB side, A1A_1 and A2A_2 on BCBC, B1B_1 and B2B_2 on ACAC). Prove that if line A1B1A_1B_1 is parallel to the bisector of angle BB, then line A2C2A_2C_2 is parallel to the bisector of angle CC.
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