  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 $T \subset
\mathbb{R}^3$ 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. 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.
