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prove A1B1C1 and A2B2C2 are similar

Source: Sharygin Finals 2023 8.2

August 2, 2023
geometrySharygin Geometry OlympiadSharygin 2023similar triangles

Problem Statement

The bisectors of angles AA, BB, and CC of triangle ABCABC meet for the second time its circumcircle at points A1A_1, B1B_1, C1C_1 respectively. Let A2A_2, B2B_2, C2C_2 be the midpoints of segments AA1AA_1, BB1BB_1, CC1CC_1 respectively. Prove that the triangles A1B1C1A_1B_1C_1 and A2B2C2A_2B_2C_2 are similar.
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