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6 points lie on surface of sphere

Source: 1958 polish Mo Finals p3

August 27, 2024
geometryspheretetrahedron3D geometry

Problem Statement

In a tetrahedron ABCD ABCD the edges ADAD, BD BD , CD CD are equal. ABC ABC Non-collinear points are chosen in the plane. A1 A_1, B1B_1, C1C_1 The lines DA1DA_1, DB1DB_1, DC1DC_1 intersect the surface of the sphere circumscribed about the tetrahedron at points A2 A_2, B2B_2, C2C_2 , different from the point D D . Prove that the points A1A_1, B1B_1, C1C_1, A2A_2, B2B_2, C2C_2 lie on the surface of a certain sphere.
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