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Four points on AP

Source: Turkey National Mathematical Olympiad 2020 P2

March 8, 2021
geometry

Problem Statement

Let PP be an interior point of acute triangle ΔABC\Delta ABC, which is different from the orthocenter. Let DD and EE be the feet of altitudes from AA to BPBP and CPCP, and let FF and GG be the feet of the altitudes from PP to sides ABAB and ACAC. Denote by XX the midpoint of [AP][AP], and let the second intersection of the circumcircles of triangles ΔDFX\Delta DFX and ΔEGX\Delta EGX lie on BCBC. Prove that APAP is perpendicular to BCBC or ∠PBA=∠PCA\angle PBA = \angle PCA.
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