円内接七・八角形の「面積x半径」公式の計算について

Abstract

This paper describes computations of the relations between the circumradius R and area S of cyclic polygons given by the lengths of the sides. The classic results of Heron and Brahmagupta clearly show that the product of R and S is expressed by the lengths of the sides for triangles and cyclic quadrilaterals. In the author's previous paper (2015), the similar integrated formulae of the circumradius and the area for cyclic pentagons and hexagons were computed using elimination by resultants and factorization of polynomials. In this study, we try to compute analogous formulae for cyclic heptagons and octagons. However, we consider the method of numerical interpolation in this case, instead of elimination. As a result, we succeeded in computing the integrated formula for cyclic heptagons, where it should be a polynomial equation in z = 4SR with degree 38 and 31, 590 terms.