Analytical solution of the weighted Fermat-Torricelli problem for convex   quadrilaterals in the Euclidean plane: The case of two pairs of equal weights
  The weighted Fermat-Torricelli problem for four non-collinear points in R^2 states that: Given four non-collinear points A_1, A_2, A_3,A_4 and a positive real number (weight) B_i which correspond to each point A_i, for i = 1, 2, 3, 4, find a fifth point such that the sum of the weighted distances to these four points is min- imized. We present an analytical solution for the weighted Fermat-Torricelli problem for convex quadrilaterals in R2 for the following two cases: (a) B_1 = B_2 and B_3 = B_4, for B1 > B4 and (b) B_1 = B_3 and B_2 = B_4. 
Anastasios N. Zachos
math.OC