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concyclicity of second intersections with sphere

Source: Sharygin 2023 - P24 (Grade-11)

March 4, 2023
geometry3D geometrysphereConcyclicInversionSharygin Geometry OlympiadSharygin 2023

Problem Statement

A tetrahedron ABCDABCD is give. A line ℓ\ell meets the planes ABC,BCD,CDA,DABABC,BCD,CDA,DAB at points D0,A0,B0,C0D_0,A_0,B_0,C_0 respectively. Let PP be an arbitrary point not lying on ℓ\ell and the planes of the faces, and A1,B1,C1,D1A_1,B_1,C_1,D_1 be the second common points of lines PA0,PB0,PC0,PD0PA_0,PB_0,PC_0,PD_0 with the spheres PBCD,PCDA,PDAB,PABCPBCD,PCDA,PDAB,PABC respectively. Prove P,A1,B1,C1,D1P,A_1,B_1,C_1,D_1 lie on a circle.
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