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concurrency in a quadrangular pyramid , 5 points concyclic

Source: All-Russian MO 2002 Regional (R4) 11.2

September 17, 2024
geometry3D geometrypyramid

Problem Statement

The altitude of a quadrangular pyramid SABCDSABCD passes through the intersection point of the diagonals of its base ABCDABCD. From the tops of the base perpendiculars AA1AA_1, BB1BB_1, CC1CC_1, DD1DD_1 are dropped onto lines SCSC, SD,SD, SASA and SBSB respectively. It turned out that the points SS, A1A_1, B1B_1, C1C_1, D1D_1 are different and lie on the same sphere. Prove that lines AA1AA_1, BB1 BB_1, CC1CC_1, DD1DD_1 pass through one point.
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