Geodesic
Generation of Relations for Bicentric Polygons
Journal for geometry and graphics, Tome 17 (2013) no. 2, pp. 141-153 Cet article a éte moissonné depuis la source Heldermann Verlag

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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 Gr�bner 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},
     url = {http://geodesic.mathdoc.fr/item/JGG_2013_17_2_JGG_2013_17_2_a1/}
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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/