The weighted Fermat-Torricelli problem for tetrahedra and an “inverse” problem

We study the weighted Fermat-Torricelli problem for tetrahedra in R³ and solve an “inverse” problem by introducing a method of differentiation. The solution of the inverse problem is the main result which states that: Given the Fermat-Torricelli point A₀ with the vertices lie on four prescribed rays, find the ratios between every pair of non-negative weights of two corresponding rays such that the sum of the four non-negative weights is a constant number. An application of the inverse weighted Fermat-Torricelli problem is the strong invariance principle of the weighted Fermat-Torricelli point which gives some classes of tetrahedra that could be named “evolutionary tetrahedra”.