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Japanese Geo

Source: 2023 Japan TST p11

July 21, 2023
geometry

Problem Statement

Let ABCABC be an acute triangle and point PP lies inside the triangle (excluding the vertices on the boundary) such that lines APAP and BCBC are not orthogonal. Let XX and YY be the points symmetric to PP wrt lines ABAB and ACAC, respectively, and let ω\omega be the circumcircle of triangle AXYAXY. Point QQ lies inside triangle ABCABC (excluding the vertices on the boundary) and satisfies ∠QBC=∠CAP\angle QBC = \angle CAP, ∠QCB=∠BAP\angle QCB = \angle BAP. Line AQAQ intersects ω\omega at a point RR, distinct from AA and QQ. Prove that the circumcircles of triangles ABCABC, PQRPQR, and ω\omega have a common point.
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