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Geometrical Solution to the Fermat Problem with Arbitrary Weights

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  • Galina Jalal
  • Jakob Krarup

Abstract

The prime motivation for the present study is a famous problem, allegedly first formulated in 1643 by Fermat, and the so-called Complementary Problem (CP), proposed but incorrectly solved in 1941 by Courant and Robbins. For a given triangle, Fermat asks for a fourth point such that the sum of its Euclidean distances, each weighted by +1, to the three given points is minimized. CP differs from Fermat in that the weight associated with one of these points is −1 instead of +1. The geometrical approach suggested in 1998 by Krarup for solving CP is here extended to cover any combination of positive and negative weights associated with the vertices of a given triangle. Among the by-products are surprisingly simple correctness proofs of the geometrical constructions of Torricelli (around 1645), Cavalieri (1647), Viviani (1659), Simpson (1750), and Martelli (1998). Furthermore, alternative proofs of Ptolemy's theorem (around A.D. 150) and an observation by Heinen (1834) are provided. Copyright Kluwer Academic Publishers 2003

Suggested Citation

  • Galina Jalal & Jakob Krarup, 2003. "Geometrical Solution to the Fermat Problem with Arbitrary Weights," Annals of Operations Research, Springer, vol. 123(1), pages 67-104, October.
    • Handle: RePEc:spr:annopr:v:123:y:2003:i:1:p:67-104:10.1023/a:1026167011686
    DOI: 10.1023/A:1026167011686
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    Cited by:

    1. Bernard Monjardet, 2008. ""Mathématique Sociale" and Mathematics. A case study: Condorcet's effect and medians," Post-Print halshs-00309825, HAL.
    2. Xu, Shaofeng, 2013. "Transport economies of scale and firm location," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 337-345.

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