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Part of 1986 IMO Longlists

Problems(1)

A1A2A3A4 is a regular tetrahedron

Source:

8/28/2010
Let OO be an interior point of a tetrahedron A1A2A3A4A_1A_2A_3A_4. Let S1,S2,S3,S4 S_1, S_2, S_3, S_4 be spheres with centers A1,A2,A3,A4A_1,A_2,A_3,A_4, respectively, and let U,VU, V be spheres with centers at OO. Suppose that for i,j=1,2,3,4,i≠ji, j = 1, 2, 3, 4, i \neq j, the spheres SiS_i and SjS_j are tangent to each other at a point BijB_{ij} lying on AiAjA_iA_j . Suppose also that UU is tangent to all edges AiAjA_iA_j and VV is tangent to the spheres S1,S2,S3,S4 S_1, S_2, S_3, S_4. Prove that A1A2A3A4A_1A_2A_3A_4 is a regular tetrahedron.
geometry3D geometrytetrahedronspheregeometry unsolved