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

Krarup, Jakob; Vajda, Steven

IMA J. Math. Appl. Bus. Ind. 8, No. 3, 215-224 (1997).

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.

MSC:

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

Keywords:

triangle; minimal distance