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Intrinsic proofs for area and circumradius of cyclic hexagons. Solving equations for arbitrary cyclic polygons

Abstract. Finding formulas for the area or circumradius of polygons inscribed in a circle in terms of side lengths is a classical subject . For the area of a triangle we have famous Heron formula and for cyclic quadrilaterals we have Brahmagupta’ s formula. A decade ago D.P.Robbins found a minimal equations satisfied by the area of cyclic pentagons and hexagons by a method of undetermined coefficients and he wrote the result in a nice compact form. The method he used could hardly be used for heptagons due to computational complexity of the approach. In another approach with two collaborators (see Ref.2) a concise heptagon/octagon area formula was obtained recently (not long after D.P.Robbins premature death). This approach uses covariants of binary quintics. It is not clear if this approach could be effectively used for cyclic polygons with nine or more sides. A nice survey on this and other Robbins conjectures is written by I.Pak (see Ref.4). In this talk we shall present an intrinsic proof of the Robbins formula for the area (circumradius and area times circumradius) of cyclic hexagon based on an intricate direct elimination of diagonals (the case of pentagon was treated in Ref.5) and using a new algorithm from Ref.6. In the early stage we used computations with MAPLE (which sometimes lasted several days!). Next we shall explain a simple quadratic system, which seems to be new, for the circumradius and area of arbitrary cyclic polygons based on a Wiener-Hopf factorization of a new Laurent polynomial invariant of cyclic polygons. Explicit formulas for the circumradius (and less explicit for the area) of cyclic heptagons and cyclic octagons are obtained. We hope to apply recent new resultant formulas of Eisenbud et al. in our approach to cyclic polygons. Acknowledgements. We would like to thank Darko Veljan and Vladimir Volenec for helpful discussions. References 1. A.F. Moebius, Ueber die Gleichungen, mittelst welcher aus der Seiten eines in einen Kreis zu beschriebenden Vielecks der Halbmesser des Kreises un die Flahe des Vielecks gefunden werden}; ; , Crelle's Journal, 3:5--34. 1828. 2. F.Miller Maley, David P.Robbins, Julie Roskies, On the areas of cyclic and semicyclic polygons, math.MG/0407300v1. 3. D.P.Robbins, Areas of polygons inscribed in a circle, Discrete Computational Geometry, 12:223--236, 1994. 4. I. Pak, The area of cyclic polygons: recent progress on Robbins conjectures, Adv. Applied Math. (to appear) 5. D. Svrtan, D.Veljan and V.Volenec, Geometry of pentagons: from Gauss to Robbins, math.MG/0403503. 6. D.Svrtan, A new approach to rationalization of surds, submitted. 7. D.Svrtan, Intrinsic proof of Robbins formula for the area of cyclic hexagons, . submitted. 8. D.Svrtan, Equations for the Circumradius and Area of cyclic polygons via Wiener-Hopf factorization. Computational aspects and some new formulas. In preparation.

circumradius; area; cyclic pentagons and hexagons; intrinsic proof

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