Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T10:10:17.207Z Has data issue: false hasContentIssue false
  • English
  • Français

The Fermat-Torricelli problem once more

Published online by Cambridge University Press:  01 August 2016

Folke Eriksson
Folke Eriksson*
Affiliation:
Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
Get access Rights & Permissions [Opens in a new window]

Extract

This extremum problem is really a classical beauty. It has had a long and interesting history since it was formulated by Fermat in the 17th century. Given three points A, B and C, the task is to find a point P such that the sum of distances PA + PB + PC is minimal; (see Figure 1). After a few years Torricelli found the solution: P should be situated so that the angles between the half-lines PA, PB and PC are all 120° (except when one angle of the triangle ABC is greater than or equal to 120°). The solution has then been rediscovered many times in new and interesting ways.

Type
Articles
Information
The Mathematical Gazette , Volume 81 , Issue 490 , March 1997 , pp. 37 - 44
Copyright
Copyright © The Mathematical Association 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Lindelöf, L. L., Sur les maxima et minima d’une fonction des rayons vecteurs menés d’un point mobile à plusieurs centres fixes, Acta Soc. Se. Fenn. 8 (1867) pp. 191–203.Google Scholar
2. Sturm, Rudolf, Ueber den Punkt kleinster Entfernungssumme von gegebenen Punkten, Jour. f. die reine u. angew. Math. (Crelle) 97 (1884) pp. 49–61.CrossRefGoogle Scholar
3. Abu-Abbas, Z., Hajja, M., A note on the Fermat point of a tetrahedron, Math. Gaz. 79 (March 1995) pp. 117–118.CrossRefGoogle Scholar
4. Eriksson, Folke, Fermat-Torricellis problem—en klassisk skönhet i delvis ny dräkt, Normal 39:2 (1991) pp. 64–75.Google Scholar
5. Eriksson, Folke, On the measure of solid angles, Math. Magazine 63 (1990) pp. 184–187.CrossRefGoogle Scholar
6. Torricelli, Evangelista, Opere 1:2, pp. 90–97.Google Scholar
7. Simpson, Thomas, Doctrine and applications of fluxions, London (1750).Google Scholar
8. Honsberger, Ross, Mathematical gems I, MAA (1973) pp. 24–34.Google Scholar
9. Steiner, Jacob, Gesammelte Werke 2, Berlin (1882) p. 729.Google Scholar
10. Courant, R., Robbins, H., What is mathematics? London, New York, Toronto (1941).Google Scholar
11. Hofmann, J. E., Elementare Lösung einer Minimumsaufgabe. Zeitschr.f. math, und naturw. Unterricht 60 (1929) pp. 22–23.Google Scholar
12. Kuhn, Harold, ‘Steiner’s’ problem revisited, Studies in optimization (ed. Dantzig-Eaves, ), MAA (1974) pp. 52–70.Google ScholarPubMed
13. Weiszfeld, E., Sur le point pour lequel la somme des distances de n points donnés est minimum, Tohoku Math. J. 43 (1937) pp. 355–386.Google Scholar