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Journal for Geometry and Graphics 17 (2013), No. 2, 141--153
Copyright Heldermann Verlag 2013

Generation of Relations for Bicentric Polygons

Alasdair McAndrew
College of Engineering and Science, Victoria University, P.O.Box 14428, Melbourne, Vic. 8001, Australia

A bicentric polygon is one which is simultaneously cyclic: all vertices lie on a circle, and tangential: all sides are simultaneously tangential to another circle. All triangles and regular polygons are trivially bicentric. In the late 18-th century, Leonhard Euler developed a formula which linked the radii R and r of the circumcircles and incircles of a triangle, and the distance d between their centres: R2-d2 = 2Rr. Shortly after, Euler's secretary, Nicolaus Fuss, managed to develop similar formulas for bicentric polygons of orders 4 to 9; these formulas have been given in many different forms subsequently. The purpose of this paper is to demonstrate how such relations can be generated by using polynomial ideals and Gröbner bases, in a manner which can be easily implemented on any modern computer algebra system.

Keywords: Bicentric polygon, Groebner bases of polynomial ideals.

MSC: 51N20; 51N35, 13P10, 68W30

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