A reformulation of the three circles theorem of Johnson [10] 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 tetra- hedron T ⊂ R 3 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 [11]. 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.