Question

证明:设G为△ABC的重心,则GA^2+GB^2+GC^2最小

Answer 1

设△ABC三点坐标分别是(x1,y1)(x2,y2),(x3,y3),G(x,y)

则

GA^2+GB^2+GC^2

=(x-x1)^2+(y-y1)^2+(x-x2)^2+(y-y2)^2+(x-x3)^2+(y-y3)^2

=3x^2-2(x1+x2+x3)+(x1^2+x2^2+x3^3)+3y^2-2(y1+y2+y3)+(y1^2+y2^2+y3^3)

根据二次函数性质,要使上式取最小值,需要

x=(x1+x2+x3)/3,y=(y1+y2+y3)/3

即G为△ABC的重心时,GA^2+GB^2+GC^2最小