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                <term xml:lang="en">Tetrahedra</term>
                <term xml:lang="en">triangular pyramids</term>
                <term xml:lang="en">three circles theorem</term>
                <term xml:lang="en">R-bodies</term>
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              <p>A reformulation of the three circles theorem of Johnson [10] with distance coordinates to thevertices of a triangle is explicitly represented in a polynomial system and solved by symboliccomputation. 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 verticesof T is an r-body [11]. For triangular pyramids we completely describe the set of geometricconfigurations with the required four balls of radius R ∗ . The solutions obtained by symboliccomputation show that triangular pyramids are splitted into two different classes: in the firstone R ∗ is unique, in the second one three values R ∗ there exist. The first class can be itselfsubdivided into two subclasses, one of which is related to the family of r-bodies.</p>
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