# Fermat point

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Re: [HM] On-line Article: Fermat Point
Re: [HM] On-line Article: Fermat Point
Harold W Kuhn (kuhn@math.Princeton.edu)                                                                                                                                                                                                                                                        Harold W Kuhn (kuhn@math.Princeton.edu)

## first appearance of Duality was a Euclidean distance problem

Rendering of Fermat point in acrylic on canvas by Suman Vaze
```Re: [HM] On-line Article: Fermat Point
Harold W Kuhn (kuhn@math.Princeton.edu)
Fri, 24 Sep 1999 21:57:28 -0400 (EDT)

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Re: [HM] On-line Article: Fermat Point
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It is always a pleasure to chat with my colleague John Conway, if only by email. We talked a bit in the elevator of Fine Hall (the building housing the Mathematics Department of Princeton University) today but he was on his way to teach a class so this must continue the conversation in public.

John Conway wrote in HM-digest 195:

As in some better-known cases, the result was conjectured by Fermat rather than proved by him. The first recorded proof is by his student, Torricelli (the inventor of the barometer). I can't remember what this was, but might be able to find my notes about it. I do recall that I didn't regard it as a very good proof, so doubt that it was the type of proof you've obviously rediscovered, which is one of the standard ones. There are indeed many references, but I don't have them off by heart; you can ask me again if you don't find any recent ones from someone else (mine are all old ones, and might not be easily accessible to you).

My article, [Kuhn,] On a Pair of Dual Nonlinear Programs (Chapter III of "Nonlinear Programming", edited by J. Abadie, North-Holland Publishing, 1967, pp.37-54), contains a historical sketch, from which I shall quote:

"Our story starts with a problem rather casually posed by Fermat early in the 17th century. At the end of his celebrated essay on maxima and minima, in which he presented pre-calculus rules for finding tangents to a variety of curves, he threw out the challenge:

[Primal] Let he who does not approve of my method attempt the solution of the following problem: Given three points in the plane, find a fourth point such that the sum of its distances to the three given points is a minimum!

The problem may have travelled to Italy with Mersenne; it is known that before 1640 Torricelli had solved the problem. He asserted that the circles circumscribing the equilateral triangles constructed on the sides of and outside the given triangle intersect in the point that is sought. This point is called the Torricelli point. Also, in Cavalieri's "Exercitationes geometricae" of 1647, it is shown that the sides of the given triangle subtend angles of 120 degrees from the Torricelli point. Furthermore, Simpson asserted and proved in his "Doctrine and Application of Fluxions" (London, 1750) that the three lines joining the outside vertices of the equilateral triangles defined above to the opposite vertices of the given triangle intersect in the Torricelli point. These three lines are called Simpson lines."

For original sources, see the article by M. Zacharias in the Encyclopaedie der Mathematischen Wissenschaften, III AB 9.

My article then erroneously gives credit to a 19th century mathematician for the original formulation of the dual problem. Further search led to earlier sources. In a remarkable journal, not much read today, "The Ladies Diary or Woman's Almanack" (1755), the following problem is posed by a Mr. Tho. Moss (p.47):

[Dual 1] In the three Sides of an equiangular Field stand three Trees, at the Distances of 10, 12, and 16 Chains from one another; to find the Content of the Field, it being the greatest the Data will admit of.

While there seems to have been no explicit recognition of the connection with Fermat's problem in the Ladies Diary, the observation was not long in coming. In the Annales de Mathematiques Pures et Appliques, edited by J. D. Gergonne, Vol.I (1810-11), we find the following problem posed on p.384:

[Dual 2] Given any triangle, circumscribe the largest possible equilateral triangle about it.

In the solution proposed by Rochat, Fauguier, and Pilatte in Vol.II (1811-12), pp.88-93, the observation is made: "Thus the largest equilateral triangle circumscribing the given triangle has sides perpendicular to the lines joining the vertices of the given triangle to the point such that the sum of the distances to these vertices is a minimum [p.91]. One can conclude that the altitude of the largest equilateral triangle that can be circumscribed about a given triangle has a length that is equal to the sum of distances from the vertices of the given triangle to the point at which the sum of distances is a minimum. [p.92]" The credit for recognizing this duality appears to be due to Vecten, professor of mathematiques speciales at the Licee de Nimes.

My own interest in Fermat's Problem originated in the computational aspects and its application to locational problems (see Kuhn, H. W., and Kuenne, R. E., J. Regional Sci. 4 (1962) 21-33, and the literature cited there). We present an efficient algorithm for solving "General Fermat Problems"; however, my paper of 1967 cited at the beginning of this note contains a proof of the duality theorem by elementary geometrical means, which may be more of interest to the students whose query started this exchange.

I cannot end this historical sketch without mention of the fact that the Fermat problem has been widely popularized by Courant and Robbins (in "What is Mathematics?") under the name of the "Steiner Problem". Although this gifted geometer of the 19th century can be counted among the dozens of mathematicians who have written on the subject, he does not seem to have contributed anything new, either to its formulation or its solution. As for the statement by Courant and Robbins that the generalization of the problem to more than three points is a sterile generalization, their answer is found in the recent literature, which has added new applications and understanding through this "sterile" extension of the problem.

See what happens, when you have a conversation in an elevator John?

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