The weighted Fermat–Torricelli–Menger problem for a given sextuple of edge lengths determining tetrahedra

We introduce the weighted Fermat–Torricelli–Menger problem for a given sextuple of edge lengths in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{3}$$\end{document} which states that: given a sextuple of edge lengths determining tetrahedra and a positive real number (weight) which corresponds to each vertex of every derived tetrahedron find the corresponding weighted Fermat–Torricelli point of these tetrahedra. We obtain a system of three rational equations with respect to three variable distances from the weighted Fermat–Torricelli point to the three vertices of the tetrahedron determined by a given sextuple of edge lengths in the sense of Menger. This system of equations gives a necessary condition to locate the weighted Fermat–Torricelli point at the interior of this class of tetrahedra and allow us to compute the position of the corresponding weighted Fermat–Torricelli point. Furthermore, we give an analytical solution for the weighted Fermat–Torricelli–Menger problem for a given sextuple of equal edge lengths in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{3}$$\end{document} for the case of two pairs of equal weights.