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3 spheres intersect alone one circle

Source: VI Soros Olympiad 1990-00 R3 11.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 28, 2024
geometry3D geometrysphereSpheres

Problem Statement

Three spheres s1s_1, s2s_2, s3s_3 intersect along one circle ω\omega. Let AA be an arbitrary point lying on the circle ω\omega. Ray ABAB intersects spheres s1s_1, s2s_2, s3s_3 at points B1B_1, B2B_2, B3B_3, respectively, ray ACAC intersects spheres s1s_1, s2s_2, s3s_3 at points C1C_1, C2C_2, C3C_3, respectively (Bi≠AiB_i \ne A_i, Ci≠AiC_i \ne A_i, i=1,2,3i=1,2,3). It is known that B2B_2 is the midpoint of the segment B1B3B_1B_3. Prove that C2C_2 is the midpoint of the segment C1C3C_1C_3.
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