Let A1A2A3A4 be a tetrahedron. We construct four mutually tangent spheres S1,S2,S3,S4 with centers A1,A2,A3,A4 respectively. Suppose that there exists a point Q such that we can construct two spheres centered at Q satisfying the following conditions:
i) One sphere with radius r is tangent to S1,S2,S3,S4;
ii) One sphere with radius R is tangent to every edges of tetrahedron A1A2A3A4.
Prove that A1A2A3A4 is a regular tetrahedron. geometry3D geometryspheretetrahedrongeometry unsolved