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Two tangent circles in an orthocenter configuration

Source: Oral Moscow geometry olympiad 2023 10-11.5

April 20, 2023
geometry

Problem Statement

In an acute-angled triangle ABCABC with orthocenter HH, the line AHAH cuts BCBC at point A1A_1. Let Γ\Gamma be a circle centered on side ABAB tangent to AA1AA_1 at point HH. Prove that Γ\Gamma is tangent to the circumscribed circle of triangle AMA1AMA_1, where MM is the midpoint of ACAC.
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