Analytical solution of the weighted Fermat-Torricelli problem for tetrahedra: The case of two pairs of equal weights

The weighted Fermat-Torricelli problem for four non-collinear and non-coplanar points in the three dimensional Euclidean Space states that: Given four non-collinear and non-coplanar points A1, A2, A3, A4 and a positive real number (weight) Bi which correspond to each point Ai, for i = 1,2,3,4, find a fifth point such that the sum of the weighted distances to these four points is minimized. We present an analytical solution for the weighted Fermat-Torricelli problem for tetrahedra in the three dimensional Euclidean Space for the case of two pairs of equal weights.