International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 283846, 16 pages
doi:10.1155/2008/283846
Research Article
The Weighted Fermat Triangle Problem
Natural Sciences and Mathematics, The Richard Stockton College of New Jersey, Pomona, NJ 08240, USA
Received 29 June 2007; Accepted 13 September 2007
Academic Editor: Marco Squassina
Copyright © 2008 Yujin Shen and Juan Tolosa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We completely solve the generalized Fermat problem: given a triangle P1, P2, P3 and
three positive numbers λ1, λ2, λ3, find a point P for which the sum λ1P1P+λ2P2P+λ3P3P
is minimal. We show that the point always exists and is unique, and indicate necessary
and sufficient conditions for the point to lie inside the triangle. We provide geometric
interpretations of the conditions and briefly indicate a connection with dynamical systems.