International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 283846, 16 pages
Research Article

The Weighted Fermat Triangle Problem

Yujin Shen and Juan Tolosa

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.


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.