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touch points adn equality of BM, M is midpoint

Source: Azerbaijan NMO 2023. Senior P5

August 24, 2023
geometryAZE SENIOR NATIONAL MO

Problem Statement

The incircle of the acute-angled triangle ABCABC is tangent to the sides AB,BC,CAAB, BC, CA at points C1,A1,B1,C_1, A_1, B_1, respectively, and II is the incenter. Let the midpoint of side BCBC be M.M. Let JJ be the foot of the altitude drawn from MM to C1B1.C_1B_1. The tangent drawn from BB to the circumcircle of △BIC\triangle BIC intersects IJIJ at X.X. If the circumcircle of △AXI\triangle AXI intersects ABAB at Y,Y, prove that BY=BM.BY = BM.
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