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Intrinsic Geometry of Cyclic Heptagons and Octagons

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, nowdays several hours!). Based on our new intermediate Brahmagupta formula we get simpler systems of equations for the area and area times circumradius of cyclic heptagons and cyclic octagons.Also computations of all diagonals will be discussed (cf..Ref 10) .It seems remarkable that our approach , with a help of Groebner basis technics leads to minimal equations , what is not the case with iterated resultants approach

cyclic heptagon; cyclic octagon; circumradius; area; diagonal; groebner basis

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