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on a sphere

Source: All Russian 2014 Grade 11 Day 2 P2

April 30, 2014
geometry3D geometryspherepyramidsymmetrycircumcircleparallelogram

Problem Statement

The sphere ω \omega passes through the vertex SS of the pyramid SABCSABC and intersects with the edges SA,SB,SCSA,SB,SC at A1,B1,C1A_1,B_1,C_1 other than SS. The sphere Ω \Omega is the circumsphere of the pyramid SABCSABC and intersects with ω \omega circumferential, lies on a plane which parallel to the plane (ABC)(ABC). Points A2,B2,C2A_2,B_2,C_2 are symmetry points of the points A1,B1,C1A_1,B_1,C_1 respect to midpoints of the edges SA,SB,SCSA,SB,SC respectively. Prove that the points AA, BB, CC, A2A_2, B2B_2, and C2C_2 lie on a sphere.
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