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文[1]将一个定值命题的系列研究([2]～[4])的结论进一步推广至多边形和多面体中,即有命题1在同一平面内的线段A1B1,A2B2,…AnBn相交于O,且均被点O平分.以O为圆心的圆半径为r,则该圆上任一点与多边形A1A2…AnB1B2…Bn各顶点连线段长度的平方和为定值.命题2不在同一平面内的线段A1B1,A2B2,…,AnBn相交于O,且均被点O平分.以O为球心的球半径为r,则该球面上任一点与多面体A1A2…AnB1B2…Bn各顶点连线段长度的平方和为定值.
In [1], the conclusion of a series of fixed value propositions ([2] ~ [4]) is further extended to polygons and polyhedrons, that is, proposition 1 segments A1B1, A2B2, ... AnBn in the same plane intersect at O , And are evenly divided by point O. The radius of the circle with O as the center is r, then the sum of the squares of the lengths of the connecting vertices of each vertex of the circle with the vertices of the polygons A1A2 ... AnB1B2 ... Bn is a fixed value. Proposition 2 is not in the same plane Lines A1B1, A2B2, ..., AnBn intersect at O and are equally divided by point O. The radius of the sphere with O as the center is r, then the square of the length of the line connecting the vertices of the polyhedron A1A2 ... AnB1B2 ... Bn And for the value.