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circumsphere lies entirely inside another circumsphere, tetrahedra related

Source: Sharygin 2005 XI CR 20

August 19, 2019
3D geometrytetrahedronspheregeometry

Problem Statement

Let II be the center of the sphere inscribed in the tetrahedron ABCD,A′,B′,C′,D′ABCD, A ', B', C ', D' be the centers of the spheres circumscribed around the tetrahedra IBCD,ICDA,IDAB,IABCIBCD, ICDA, IDAB, IABC, respectively. Prove that the sphere circumscribed around ABCDABCD lies entirely inside the circumscribed around A′B′C′D′A'B'C'D '.
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