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Tangent points of a quadrilateral to a sphere are coplanar

Source: Spanish Mathematical Olympiad, National Stage, 2011

August 9, 2011
geometry3D geometryspheregeometry unsolved

Problem Statement

Let AA, BB, CC, DD be four points in space not all lying on the same plane. The segments ABAB, BCBC, CDCD, and DADA are tangent to the same sphere. Prove that their four points of tangency are coplanar.