Svrtanによるnew Brahmagupta's formulaの円内接多角形問題への適用について

Abstract

This paper describes computations of the relations between the circurnradius R and area S of cyclic polygons given by the lengths of the sides. The classic results of Heron and Brahrnagupta 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 contrast, we revisit the computation of pentagon and hexagon formulae applying “new Brahmagupta's formula” discovered by Svrtan. As a result, we succeeded in computing the integrated formulae for cyclic pentagons and hexagons more efficiently. Analogously, we tried to apply “new Brahmagupta's formula” for the heptagon and octagon cases. However, the elimination process has remained still very difficult, and there seems no other applicable methods than the method of numerical interpolation.