The Weighted Fermat Triangle Problem

الملخص EN

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.