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prove all lines A'B' concur

Source: Sharygin Finals 2023 9.3

August 2, 2023
geometrySharygin Geometry OlympiadSharygin 2023concurrent lines

Problem Statement

Points A1A_1, A2A_2, B1B_1, B2B_2 lie on the circumcircle of a triangle ABCABC in such a way that A1B1∥ABA_1B_1 \parallel AB, A1A2∥BCA_1A_2 \parallel BC, B1B2∥ACB_1B_2 \parallel AC. The line AA2AA_2 and CA1CA_1 meet at point A′A', and the lines BB2BB_2 and CB1CB_1 meet at point B′B'. Prove that all lines A′B′A'B' concur.
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