A Fermat point of a triangle minimizes the sum of Euclidean distances to its three vertices. If every interior angle is less than , the minimizer lies inside, and the three joining segments meet at pairwise angles of . Indeed the gradient of the sum at an interior minimizer is the sum of three unit vectors toward the vertices; three unit vectors sum to zero exactly in this configuration. Convexity of distance makes this stationary point a global minimizer. If one interior angle is at least , that vertex is a minimizer instead. For an equilateral triangle, the Fermat point is its centroid.
New to topics? Read the docs here!