Computations of the Area and Radius of Cyclic Polygons Given by the Lengths of Sides

  • Pavel Pech
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3763)


Some properties of inscribed polygons, i.e., such plane polygons whose vertices lie on a circle, are investigated. Given an inscribed polygon with the lengths of its sides, we explore the area and radius of its circumcircle. We start with a triangle and a quadrangle and then we will explore the case of a pentagon. All the computations are based on results of commutative algebra especially on Gröbner bases method and elimination of variables in a given ideal.


Theorem Prove Convex Case Elementary Symmetric Function Coordinate Method Geometry Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berger, M.: Geometry I. Springer, Heidelberg (1987)zbMATHCrossRefGoogle Scholar
  2. 2.
    Blaschke, W.: Kreis und Kugel. Walter de Gruyter & Co., Berlin (1956)Google Scholar
  3. 3.
    Chou, S.-C.: Mechanical Geometry Theorem Proving. D. Reidel Publishing Company, Dordrecht (1987)Google Scholar
  4. 4.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 2nd edn. Springer, New York (1997)Google Scholar
  5. 5.
    Coxeter, H.S.M., Greitzer, S.L.: Geometry Revisited, Toronto New York (1967)Google Scholar
  6. 6.
    Dörrie, B.H.: Triumph der Mathematik. Breslau (1933)Google Scholar
  7. 7.
    Kowalewski, G.: Einfürung in die Determinantentheorie. Veit & Comp., Leipzig (1909)Google Scholar
  8. 8.
    Pak, I.: The Area of Cyclic Polygons: Recent Progress on Robbins’ Conjectures. Mini Survey, 5 pages to appear in Adv. Applied Math (special issue in memory of David Robbins)Google Scholar
  9. 9.
    Rashid, M.A., Ajibade, A.O.: Two Conditions for a Quadrilateral to be Cyclic Expressed in Terms of the Lengths of Its Sides. Int. J. Math. Educ. Sci. Techn. 34(5), 739–742 (2003)CrossRefGoogle Scholar
  10. 10.
    Recio, T., Sterk, H., Vélez, M.P.: Project 1. Automatic Geometry Theorem Proving. In: Some Tapas of Computer Algebra. In: Cohen, A., Cuipers, H., Sterk, H. (eds.) Algorithms and Computations in Mathematics, vol. 4, pp. 276–296. Springer, Heidelberg (1998)Google Scholar
  11. 11.
    Rédey, L., Nagy, B.S.Z.: Eine Verallgemeinerung der Inhaltsformel von Heron. Publ. Math. Debrecen 1, 42–50 (1949)MathSciNetGoogle Scholar
  12. 12.
    Robbins, D.P.: Areas of Polygons Inscribed in a Circle. Discrete Comput. Geom. 12, 223–236 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Sadov, S.: Sadov’s Cubic Analog of Ptolemy’s Theorem,
  14. 14.
    Schreiber, P.: On the Existence and Constructibility of Inscribed Polygons. Beiträge zur Algebra und Geometrie 34, 195–199 (1993)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Staudt, C.R.: Über die Inhalte der Polygone und Polyeder. Journal für die reine und angewandte Mathematik 24, 252–256 (1842)zbMATHCrossRefGoogle Scholar
  16. 16.
    Svrtan, D., Veljan, D., Volenec, V.: Geometry of Pentagons: From Gauss to Robbins,
  17. 17.
    Wang, D.: Gröbner Bases Applied to Geometric Theorem Proving and Discovering. In: Buchberger, B., Winkler, F. (eds.) Gröbner Bases and Applications, pp. 281–301. Cambridge University Press, Cambridge (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Pavel Pech
    • 1
  1. 1.University of South BohemiaBudějoviceCzech Republic

Personalised recommendations