(x+y)^2+x^2+y^2>0
→3（x^2+y^2）+2xy>0
→4（x^2+y^2）+4xy>x^2+y^2+2xy=(x+y)^2
→x^2+y^2+xy>(1/2(x+y))^2
→sqr(x~2+xy+y~2)>1/2(x+y)
同理sqr(y~2+zy+z~2)>1/2(z+y)
sqr(z~2+xz+x~2)>1/2(z+x)
三式相加 得sqr(x~2+xy+y~2)+sqr(y~2+zy+z~2)+sqr(z~2+xz+x~2)>3/2(x+y+z)
证毕
→3（x^2+y^2）+2xy>0
→4（x^2+y^2）+4xy>x^2+y^2+2xy=(x+y)^2
→x^2+y^2+xy>(1/2(x+y))^2
→sqr(x~2+xy+y~2)>1/2(x+y)
同理sqr(y~2+zy+z~2)>1/2(z+y)
sqr(z~2+xz+x~2)>1/2(z+x)
三式相加 得sqr(x~2+xy+y~2)+sqr(y~2+zy+z~2)+sqr(z~2+xz+x~2)>3/2(x+y+z)
证毕
2楼2005-08-04 14:20收起回复
