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Lines are concur, equilateral triangles, square

Source: Oral Moscow Geometry Olympiad 2024, 8-9.3

September 3, 2024
geometry

Problem Statement

An equilateral triangle ABEABE is built inside the square ABCDABCD on the side ABAB, and an equilateral triangle AFCAFC is built on the diagonal ACAC (DD is inside this triangle). The segment EFEF intersects CDCD at point PP. Prove that the lines APAP, BEBE and CFCF intersect at the same point.