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4 intersection points lie on a circle coaxial with two given circles

Source: 2021 Sharygin Geometry Olympiad Finals grades X-XI p5

August 2, 2021
Concycliccoaxial circlesgeometry

Problem Statement

A secant meets one circle at points A1A_1, B1B_1։, this secant meets a second circle at points A2A_2, B2B_2. Another secant meets the first circle at points C1C_1, D1D_1 and meets the second circle at points C2C_2, D2D_2. Prove that point A1C1∩B2D2A_1C_1 \cap B_2D_2, A1C1∩A2C2A_1C_1 \cap A_2C_2, A2C2∩B1D1A_2C_2 \cap B_1D_1, B2D2∩B1D1B_2D_2 \cap B_1D_1 lie on a circle coaxial with two given circles.
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