On the configurations of four spheres supporting the vertices of a tetrahedron

A reformulation of the three circles theorem of Johnson with distance coordinates to the vertices of a triangle is explicitly represented in a polynomial system and solved by symbolic computation. A similar polynomial system in distance coordinates to the vertices of a tetrahedron T R³ is introduced to represent the configurations of four spheres of radius R^*, which intersect in one point, each sphere containing three vertices of T but not the fourth one. This problem is related to that of computing the largest value r for which the set of vertices of T is an r-body. For triangular pyramids we completely describe the set of geometric configurations with the required four balls of radius R^*. The solutions obtained by symbolic computation show that triangular pyramids are splitted into two different classes: in the first one R^* is unique, in the second one three values R^* there exist. The first class can be itself subdivided into two subclasses, one of which is related to the family of r-bodies.