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SO _|_ (A_1B_1C), regular triangular pyramid SABC , sphere ( A,B,C_1,S)

Source: All - Russian Regional Olympiad (Round 4) 1994 11.7

August 26, 2024
geometry3D geometrypyramidsphere

Problem Statement

Points A1A_1, B1B_1 and C1C_1 are taken on the respective edges SASA, SBSB, SCSC of a regular triangular pyramid SABCSABC so that the planes A1B1C1A_1B_1C_1 and ABCABC are parallel. Let OO be the center of the sphere passing through AA, BB, C1C_1 and SS. Prove that the line SOSO is perpendicular to the plane A1B1CA_1B_1C.
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