已知x+y+z=xyz,证明:x(1-y2)(1-z2)+y(1-x2)(1-z2)+z(1-x2)(1-y2)=4xy
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已知x+y+z=xyz,证明:x(1-y2)(1-z2)+y(1-x2)(1-z2)+z(1-x2)(1-y2)=4xyz.
证明:∵x+y+z=xyz,
∴左边=x(1-z2-y2-y2z2)+y(1-z2-x2+x2z2)+(1-y2-x2+x2y2)
=(x+y+z)-xz2-xy2+xy2z2-yz2+yx2+yx2z2-zy2-zx2+zx2y2
=xyz-xy(y+x)-xz(x+z)-yz(y+z)+xyz(xy+yz+zx)
=xyz-xy(xyz-z)-xz(xyz-y)-yz(xyz-x)+xyz(xy+yz+zx)
=xyz+xyz+xyz+xyz
=4xyz=右边.
∴x(1-y2)(1-z2)+y(1-x2)(1-z2)+z(1-x2)(1-y2)=4xyz.
∴左边=x(1-z2-y2-y2z2)+y(1-z2-x2+x2z2)+(1-y2-x2+x2y2)
=(x+y+z)-xz2-xy2+xy2z2-yz2+yx2+yx2z2-zy2-zx2+zx2y2
=xyz-xy(y+x)-xz(x+z)-yz(y+z)+xyz(xy+yz+zx)
=xyz-xy(xyz-z)-xz(xyz-y)-yz(xyz-x)+xyz(xy+yz+zx)
=xyz+xyz+xyz+xyz
=4xyz=右边.
∴x(1-y2)(1-z2)+y(1-x2)(1-z2)+z(1-x2)(1-y2)=4xyz.
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