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one of XB_1, OA_1 and C_1Y is equal to sum of other two

Source: 2022 Oral Moscow Geometry Olympiad grades 10-11 p1

April 18, 2022
geometryequal segments

Problem Statement

In a circle with center OO, chords ABAB and ACAC are drawn, both equal to the radius. Points A1A_1, B1B_1 and C1C_1 are projections of points A,BA, B and CC, respectively, onto an arbitrary diameter XYXY. Prove that one of the segments XB1XB_1, OA1OA_1 and C1YC_1Y is equal to the sum of the other two.
(A. Shklover)