Geometrical solution to the Fermat problem with arbitrary weights
Given three points in the plane, the Fermat problem is to find a fourth point such that the sum of (weighted) distances to the given points is minimized. After giving a detailed survey of the related literature, geometric and analytic solutions are presented for arbitrary combinations of positive and negative weights. In the case that all weights are positive, a geometric argument related to the triangle inequality plays a central role in the construction. If, on the other hand, at least one of the weights is negative, an earlier approach of J. Krarup (1998) is extended which is based on the definition of a set of ``companion instances'' with positive weights. The results imply alternative and relatively simple correctness proofs for the constructions of Torricelli (1645), Cavalieri (1647), Viviani (1659), Simpson (1750), and Martelli (1998).
scientific article; zbMATH DE number 2021309