Geodesic
Generation of Relations for Bicentric Polygons
Journal for geometry and graphics, Tome 17 (2013) no. 2, pp. 141-153
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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.
Classification : 51N20, 51N35, 13P10, 68W30
Mots-clés : Bicentric polygon, Groebner bases of polynomial ideals 
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     author = {A. McAndrew},
     title = {Generation of {Relations} for {Bicentric} {Polygons}},
     journal = {Journal for geometry and graphics},
     pages = {141--153},
     year = {2013},
     volume = {17},
     number = {2},
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A. McAndrew. Generation of Relations for Bicentric Polygons. Journal for geometry and graphics, Tome 17 (2013) no. 2, pp. 141-153. http://geodesic.mathdoc.fr/item/JGG_2013_17_2_JGG_2013_17_2_a1/