On equivalence of conditions for a quadrilateral to be cyclic

Authors:
Pavel Pech
Affiliations:
Faculty of Education, University of South Bohemia, Ceské Budějovice, Czech Republic
Venue:
ICCSA'11 Proceedings of the 2011 international conference on Computational science and its applications - Volume Part IV
Year:
2011

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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.