ILL 1986 Problem 12

Let $O$ be an interior point of a tetrahedron $A_1A_2A_3A_4$ . Let $S_1, S_2, S_3, S_4$ be spheres with centers $A_1,A_2,A_3,A_4$ , respectively, and let $U, V$ be spheres with centers at $O$ . Suppose that for $i, j = 1, 2, 3, 4, i \neq j$ , the spheres $S_i$ and $S_j$ are tangent to each other at a point $B_{ij}$ lying on $A_iA_j$ . Suppose also that $U$ is tangent to all edges $A_iA_j$ and $V$ is tangent to the spheres $S_1, S_2, S_3, S_4$ . Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.