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Locus of intersection of (A_1BC_2) and (A_2BC_1)

Source: Oral Moscow Geometry Olympiad 2024, 10-11.5

September 3, 2024
geometryparallelogram

Problem Statement

From point DD of parallelogram ABCDABCD were drawn an arbitrary line 1\ell_1, intersecting the segment ABAB and the line BCBC at points C1C_1 and A1A_1, respectively, and an arbitrary line 2\ell_2 intersecting the segment BCBC and the line ABAB at the points A2A_2 and C2C_2, respectively. Find the locus of the intersection points of the circles (A1BC2)(A_1BC_2) and (A2BC1)(A_2BC_1) (other than point BB).