# 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},
year={2005},
volume={34},
pages={669-689}
}```
• Published 16 July 2004
• Mathematics
• Adv. Appl. Math.
26 Citations

### 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

### A "right" path to cyclic polygons

• Mathematics
• 2019
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

### The geometry of cyclic hyperbolic polygons

A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if its vertices lie on a metric circle, horocycle, or a component of the equidistant locus to a hyperbolic geodesic,

### Intrinsic Geometry of Cyclic Polygons via ”New” Brahmagupta Formula

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

### Computation and Analysis of Explicit Formulae for the Circumradius of Cyclic Polygons (Extended Abstract)

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

## References

SHOWING 1-7 OF 7 REFERENCES

### Rigidity and polynomial invariants of convex polytopes

• Mathematics
• 2005
We present an algebraic approach to the classical problem of constructing a simplicial convex polytope given its planar triangulation and lengths of its edges. We introduce polynomial invariants of a

### Discriminants, Resultants, and Multidimensional Determinants

• Mathematics
• 1994
Preface.- Introduction.- General Discriminants and Resultants.- Projective Dual Varieties and General Discriminants.- The Cayley Method of Studying Discriminants.- Associated Varieties and General

### Areas of polygons inscribed in a circle

AbstractHeron of Alexandria showed that the areaK of a triangle with sidesa,b, andc is given by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn%

### Classical Invariant Theory

Introduction Notes to the reader A brief history Acknowledgements 1. Prelude - quadratic polynomials and quadratic forms 2. Basic invariant theory for binary forms 3. Groups and transformations 4.