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Beautiful geo finale of day 1

Source: All-Russian MO 2023 Final stage 11.4

April 23, 2023
geometry

Problem Statement

Let ω\omega be the circumcircle of triangle ABCABC with AB<ACAB<AC. Let II be its incenter and let MM be the midpoint of BCBC. The foot of the perpendicular from MM to AIAI is HH. The lines MH,BI,ABMH, BI, AB form a triangle TbT_b and the lines MH,CI,ACMH, CI, AC form a triangle TcT_c. The circumcircle of TbT_b meets ω\omega at B′B' and the circumcircle of TcT_c meets ω\omega at C′C'. Prove that B′,H,C′B', H, C' are collinear.
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