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AA_1, BB_1, CC_1 concurrent in isogonal conjugate diagram + angle condition

Source: Sharygin 2019 Finals Day 1 Grade 10 P3

July 30, 2019
geometry

Problem Statement

Let PP and QQ be isogonal conjugates inside triangle ABCABC. Let ω\omega be the circumcircle of ABCABC. Let A1A_1 be a point on arc BCBC of ω\omega satisfying ∠BA1P=∠CA1Q\angle BA_1P = \angle CA_1Q. Points B1B_1 and C1C_1 are defined similarly. Prove that AA1AA_1, BB1BB_1, CC1CC_1 are concurrent.
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