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

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

Recommended by Marco Squassina

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.