  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.
