On Torricelli’s geometrical solution to a problem of Fermat. (English) Zbl 0901.51010

Summary: Around 1640, Torricelli devised a geometrical solution to a problem, allegedly first formulated in the early 1600s by Fermat: ‘given three points in a plane, find a fourth point such that the sum of its distances to the three given points is as small as possible’. We account for Torricelli’s construction together with a correctness proof which also establishes the validity of results obtained much later. We introduce furthermore a so-called complementary problem, arising when the given triangle has one angle exceeding \(120^\circ\), and for which an incorrect solution was given in 1941 by Courant & Robbins. Some historical notes conclude the paper.


51M04 Elementary problems in Euclidean geometries
90B85 Continuous location
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)