On the configurations of four spheres supporting the vertices of a tetrahedron
<|reference_start|>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 $T \subset \mathbb{R}^3$ 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.<|reference_end|>
@article{longinetti2024on,
    title={On the configurations of four spheres supporting the vertices of a
  tetrahedron},
    author={Marco Longinetti, Simone Naldi},
    journal={arXiv preprint arXiv:2405.16167},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.16167},
    primaryClass={math.MG cs.SC math.AG}
}