P. Lisonek, R. Israel,
Metric invariants of tetrahedra via polynomial elimination.
Proceedings of the 2000 International Symposium on Symbolic 
and Algebraic Manipulation (C. Traverso, Ed.), pp. 217-219. ACM Press 2000.
We use Gröbner bases and the Pedersen-Roy-Szpirglas real solution counting method to analyze systems of algebraic equations satisfied by metric invariants of the tetrahedron. The transformation from the geometric to the algebraic setting is effected using the Cayley-Menger determinant formula for the volume of a simplex. We prove that, in general, the four face areas, circumradius and volume together do not uniquely determine a tetrahedron, and that there exist non-regular tetrahedra that are uniquely determined just by the four face areas and circumradius. This settles two open problems posed by the American Mathematical Monthly in 1999.

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