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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 , and for which an incorrect solution was given in 1941 by Courant & Robbins. Some historical notes conclude the paper.
MSC:
51M04Elementary problems in Euclidean geometries
90B85Continuous location
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)