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)


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.


cyclic quadrilaterals Ptolemy inequality automated geometry theorem proving 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

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

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