Advertisement

On Equivalence of Conditions for a Quadrilateral to Be Cyclic

  • Pavel Pech
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6785)

Abstract

In the paper we will prove a theorem that puts together three conditions — Ptolemy, Cubic and Quartic — for a convex quadrilateral to be cyclic. Further Ptolemy inequality is proved. Some related formulas from geometry of polygons are derived as well. These computations were done by the theory of automated geometry theorem proving using Gröbner bases approach. Dynamic geometry system GeoGebra was applied to verify Ptolemy conditions. These conditions were subsequently proved by Wu–Ritt method using characteristic sets. The novelty of the paper is the method of proving geometric inequalities. Also some relations among Ptolemy, Cubic and Quartic conditions seem to be new.

Keywords

cyclic quadrilaterals Ptolemy inequality automated geometry theorem proving 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berger, M.: Geometry I. Springer, Heidelberg (1987)Google Scholar
  2. 2.
    Chou, S.C.: Mechanical Geometry Theorem Proving. D. Reidel Publishing Company, Dordrecht (1987)CrossRefzbMATHGoogle Scholar
  3. 3.
    Chou, S.C., Gao, X.S., Arnon, D.S.: On the mechanical proof of geometry theorems involving inequalities. Advances in Computing Research 6, 139–181 (1992)Google Scholar
  4. 4.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms. Springer, Berlin (1997)zbMATHGoogle Scholar
  5. 5.
    Kowalewski, G.: Einführung in die Determinantentheorie. Veit & Comp. Leipzig (1909)zbMATHGoogle Scholar
  6. 6.
    Pech, P.: Selected topics in geometry with classical vs. computer proving. World Scientific Publishing, New Jersey (2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    Pech, P.: On the Need of Radical Ideals in Automatic Proving: A Theorem about Regular Polygons. In: Botana, F., Recio, T. (eds.) ADG 2006. LNCS (LNAI), vol. 4869, pp. 157–170. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    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, 739–742 (2003)CrossRefGoogle Scholar
  9. 9.
    Recio, T., Vélez, M.P.: Automatic Discovery of Theorems in Elementary Geometry. J. Automat. Reason. 12, 1–22 (1998)zbMATHGoogle Scholar
  10. 10.
    Sadov, S.: Sadov’s Cubic Analog of Ptolemy’s Theorem sadov.html (2004), http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/
  11. 11.
    Wang, D.: Gröbner Bases Applied to Geometric Theorem Proving and Discovering. In: Buchberger, B., Winkler, F. (eds.) Gröbner Bases and Applications. Lecture Notes of Computer Algebra, pp. 281–301. Cambridge Univ. Press, Cambridge (1998)CrossRefGoogle Scholar
  12. 12.
    Wang, D.: Elimination Methods. Springer, Wien New York (2001)Google Scholar
  13. 13.
    Wang, D.: Elimination Practice. Software Tools and Applications. Imperial College Press, London (2004)CrossRefGoogle Scholar
  14. 14.
    Wu, W.-t.: Mathematics Mechanization. Science Press, Beijing; Kluwer Academic Publishers, Dordrecht (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Pavel Pech
    • 1
  1. 1.Faculty of EducationUniversity of South BohemiaČeské BudějoviceCzech Republic

Personalised recommendations