圆内接(外切)多边形面积的极值性质

Extremal properties on inscribed (circumscribed) polygons of a circle

关于定圆的内接n边形,本文用两种方法证明了,圆的内接正n边形面积最大.关于圆的外切多边形,本文引入了对偶多边形这一新的概念,从而得到了如下结果,在定圆的所有外切n边形中,以外切正n边形面积最小.

In this paper by two methods we prore that the area of inscribed regular n- polygons of a circle is maximal than areas of all inscribed n-polygons of that circle. On circumscribed polygons of a circle, we introduce a new concept—dual polygon. And so is obtained the following result: area of circumscribed regular n-polygons of a circle is minimal in all circumscribed n-polygons of that circle.