  We obtain two analytic solutions for the weighted Fermat-Torricelli problem
in the Euclidean Plane which states that: Given three points in the Euclidean
plane and a positive real number (weight) which correspond to each point, find
the point such that the sum of the weighted distances to these three points is
minimized. Furthermore, we give two new geometrical solutions for the weighted
Fermat-Torricelli problem (weighted Fermat-Torricelli point), by using the
floating equilibrium condition of the weighted Fermat-Torricelli problem (first
geometric solution) and a generalization of Hofmann's rotation proof under the
condition of equality of two given weights (second geometric solution).
