Integrated Circumradius and Area Formulae for Cyclic Pentagons and Hexagons
Integrated Circumradius and Area Formulae for Cyclic Pentagons and Hexagons
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DOI：
10.1007/978-3-319-21362-0_6
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发表时间：
2014-07
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影响因子：
--
通讯作者：
Shuichi Moritsugu
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文献类型：
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作者：
Shuichi Moritsugu
本文给出了由边长给出的循环多边形的外接圆半径R与面积S之间关系的计算方法。Heron和Brahmagupta的经典结果清楚地表明，对于三角形和循环四边形，Rand的乘积是由边长表示的。然而，对于五边形和六边形的外切半径和面积公式，人们一直是分别研究的，很少讨论它们之间的关系。本文在Robbins（1994），Pech（2006）和作者（2011）的基础上，成功地计算了循环五边形和六边形的外切半径和面积的积分公式。对于五边形，它们是一个7次多项式方程;对于六边形，它们是两个7次多项式的乘积。我们证明了这三个7次多项式是用交叉宇称的概念统一表示的，交叉宇称的结构类似于的面积公式和外切半径公式。此外，我们还导出了（4SR）中关于循环五边形的一个7次多项式方程，并证明了这类公式只存在于n-ε中，其中ε是奇数.
This paper describes computations of the relations between the circumradiusRand areaSof cyclic polygons given by the lengths of the sides. The classic results of Heron and Brahmagupta clearly show that the product ofRandSis expressed by the lengths of the sides for triangles and cyclic quadrilaterals. However, the formulae of circumradius and area for cyclic pentagons and hexagons have been studied separately, and the relation between them has seldom been discussed. In this study, based on the results derived by Robbins (1994), Pech (2006), and the author (2011), we succeeded in computingintegrated formulaeof the circumradius and the area for cyclic pentagons and hexagons. They are found to be a polynomial equation inwith degree 7 for pentagons, and the product of two polynomials each with degree 7 for hexagons. We confirmed that these three polynomials with degree 7 are uniformly expressed using the notion ofcrossing parity, the structure of which is analogous to those of the area formulae and circumradius formulae for. Moreover, we derived a polynomial equation in (4SR) itself with degree 7 for cyclic pentagons, and showed that this type of formula exists only forn-gons, wherenis an odd number.