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MA_2 bisects BC, circumrcircle

Source: 2020 Cono Sur Shortlist G4 https://artofproblemsolving.com/community/c1088686_cono_sur_shortlist__geometry

November 30, 2021
bisects segmentgeometrycircumcircle

Problem Statement

Let ABCABC be a triangle with circumcircle ω\omega. The bisector of ∠BAC\angle BAC intersects ω\omega at point A1A_1. Let A2A_2 be a point on the segment AA1AA_1, CA2CA_2 cuts ABAB and ω\omega at points C1C_1 and C2C_2, respectively. Similarly, BA2BA_2 cuts ACAC and ω\omega at points B1B_1 and B2B_2, respectively. Let MM be the intersection point of B1C2B_1C_2 and B2C1B_2C_1. Prove that MA2MA_2 passes the midpoint of BCBC.
proposed by Jhefferson Lopez, Perú
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