圆内接多边形定周长的面积最大值问题

Maximum Area of Inscribed Polygon with Constant Perimeter

推广了经典的圆内接三角形面积最大值问题。运用分析的方法给出了圆内接三角形定周长的面积最大的三角形为等腰三角形 ,并且给出了周长和面积最大值的不等式关系。将此结果推广到圆内接多边形的情况 ,得到了在周长为定值的条件下圆内接 n边 -多边形中 ,面积最大的 n边形最多只有两种不同的边长。

Classical maximum area of inscribed triangle is extended in this paper.It is obtained that the inscribed triangle with constant perimeter is isosceles when using mathematical analysis to maximize its area.Besides,an inequality of the maximum of area and perimeter is established,i.e. S max ≤(1/4) Rl and maximum area of inscribed polygon in a circle with constant perimeter is given by theorem 2,which shows that there exist at most two different lengths of the sides for the polygon.