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arXiv · 2307.00034

A fundamental property of the Fermat-Torricelli point for tetrahedra in the three dimensional Euclidean Space

Anastasios N. Zachos
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Abstract

We prove the following fundamental property for the Fermat-Torricelli point for four non-collinear and non-coplanar points forming a tetrahedron in $\mathbb{R}^{3},$ which states that: The three bisecting lines having as a common vertex the Fermat-Torricelli point formed by each pair of equal angles, which are seen by the opposite edges of the tetrahedron meet perpendicularly at the Fermat-Torricelli point. Furthermore, we give an alternative proof, which is different from the one obtained by Bajaj and Mehlhos for the unsolvability of the Fermat-Torricelli problem for tetrahedra in $\mathbb{R}^{3}$ using only algebraic computations for some angles, which have as a common vertex the Fermat-Torricelli point of the tetrahedron.

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Anastasios N. Zachos. 2023-06-30. A fundamental property of the Fermat-Torricelli point for tetrahedra in the three dimensional Euclidean Space. https://arxiv.org/abs/2307.00034

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