International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 283846, 16 pages
doi:10.1155/2008/283846
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
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.