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

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