On the areas of cyclic and semicyclic polygons

@article{Maley2005OnTA,
  title={On the areas of cyclic and semicyclic polygons},
  author={F. Miller Maley and David P. Robbins and Julie Roskies},
  journal={Adv. Appl. Math.},
  year={2005},
  volume={34},
  pages={669-689}
}

On Circumradius Equations of Cyclic Polygons

In a masterfully written (in german language) thirty pages long paper (and published in 1828 in Crelle’s Journal ) A. F. Möbius studied some properties of the polynomial equations for the

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It is well known that Heron's theorem provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its sides. It has been extended by Brahmagupta to

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Finding explicit equations for the area or circumradius of polygons inscribed in a circle in terms of side lengths is a classical subject (cf.[1]). For triangle / cyclic quadrilaterals we have famous

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A more efficient method for computing the circumradius of cyclic heptagons than before is found and 25 out of 39 coefficients in the ci_{1} .cumradius formula for cyclic octagons are succeeded in.

Computation and Analysis of Explicit Formulae for the Circumradius of Cyclic Polygons ∗

The present work has succeeded in explicitly computing the circumradius of cyclic heptagons, which is converted into an expression in the form of elementary symmetric polynomials for the first time.

Integrated Circumradius and Area Formulae for Cyclic Pentagons and Hexagons

Computations of the relations between the circumradius R and area S of cyclic polygons given by the lengths of the sides are described, and a polynomial equation in 4SR itself with degree 7 for cyclic pentagons is derived, showing that this type of formula exists only for n-gons, where n is an odd number.

Computation with Pentagons

The paper deals with properties of pentagons in a plane which are related to the area of a pentagon. First the formulas of Gauss and Monge holding for any pentagon in a plane are studied. Both

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