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

Two new analytical solutions and two new geometrical solutions for the weighted Fermat-Torricelli problem in the Euclidean plane