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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