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Journal for Geometry and Graphics 19 (2015), No. 1, 031--042
Copyright Heldermann Verlag 2015

Two New Analytical and Two New Geometrical Solutions for the Weighted Fermat-Torricelli Problem in the Euclidean Plane

Anastasios Zachos
Department of Mathematics, University of Patras, Greece

We obtain two analytic solutions for the weighted Fermat-Torricelli problem in the Euclidean Plane which states: 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 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).

Keywords: Weighted Fermat-Torricelli point, floating case, absorbed case, median, ruler and compass construction.

MSC: 51M04; 51M16, 51M15, 74P20

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