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Midpoints and circumcircles lead to another midpoint

Source: 2021 Iberoamerican Mathematical Olympiad, P2

October 20, 2021
geometrycircumcircle

Problem Statement

Consider an acute-angled triangle ABCABC, with AC>ABAC>AB, and let Γ\Gamma be its circumcircle. Let EE and FF be the midpoints of the sides ACAC and ABAB, respectively. The circumcircle of the triangle CEFCEF and Γ\Gamma meet at XX and CC, with X≠CX\neq C. The line BXBX and the tangent to Γ\Gamma through AA meet at YY. Let PP be the point on segment ABAB so that YP=YAYP = YA, with P≠AP\neq A, and let QQ be the point where ABAB and the parallel to BCBC through YY meet each other. Show that FF is the midpoint of PQPQ.
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