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Reflecting a circle, you find another one

Source: Baltic Way 2020, Problem 14

November 14, 2020
geometrygeometry proposed

Problem Statement

An acute triangle ABCABC is given and let HH be its orthocenter. Let ω\omega be the circle through BB, CC and HH, and let Γ\Gamma be the circle with diameter AHAH. Let X≠HX\neq H be the other intersection point of ω\omega and Γ\Gamma, and let γ\gamma be the reflection of Γ\Gamma over AXAX.
Suppose γ\gamma and ω\omega intersect again at Y≠XY\neq X, and line AHAH and ω\omega intersect again at Z≠HZ \neq H. Show that the circle through A,Y,ZA,Y,Z passes through the midpoint of segment BCBC.
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