The Weighted Fermat Triangle Problem - Yujin Shen; Juan Tolosa (International Journal of Mathematics and Mathematical Sciences)
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.