设P为正n边形A_1A_2…A_n外接圆上任意一点,R为这正n边形外接圆半径,则P到各顶点距离平方和为定值2nR~2,即 sum from i=1 to n PA_i~2=2nR~2 (1) 本文试对这一有趣的定值问题作适当引伸,得到一些更一般的结论。定理1 设正n边形A_1A_2…A_n的中心为O,半径为R,P是以O为圆心以r为半径的圆 Let P be a positive n-angle A_1A_2...A_n any point on the circumscribed circle, R is the radius of the circumscribed circle of this positive n-side rim, then the square sum of P to each vertex is a fixed value 2nR~2, ie sum from i=1 to n PA_i~2=2nR~2 (1) This paper tries to make an appropriate extension to this interesting fixed-value problem to get some more general conclusions. Theorem 1 Let the center of the positive n-polygon A_1A_2...A_n be O, the radius be R, and P be a circle with O as the radius and r as the radius.