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6 points lie on a sphere, starting with a tetrahedron

Source: 1962-63 Germany R4 12.6 https://artofproblemsolving.com/community/c3208025_

October 12, 2024
geometry3D geometryspheretetrahedron

Problem Statement

Consider a pyramid ABCDABCD whose base ABCABC is a triangle. Through a point MM of the edge DADA, the lines MNMN and MPMP on the plane of the surfaces DABDAB and DACDAC are drawn respectively, such that NN is on DBDB and PP is on DCDC and ABNMABNM , ACPMACPM are cyclic quadrilaterals. a) Prove that BCPNBCPN is also a cyclic quadrilateral. b) Prove that the points A,B,C,M,N,PA,B,C,M,N, P lie on a sphere.
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