On Torricelli's geometrical solution to a problem of Fermat

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°, and for which an incorrect solution is given in 1941 by Courant & Robbins. Some historical notes conclude the paper.