作业帮设x,y,z是实数,且(x-y)^2+(y-z)^2+(z-x)^2=(x+y-2z)^2+(y+z-2x)^2+(z+x-2y)^2求[(xy+1)(yz+1)(zx+1)]/[(x^2+1)(y^2+1)(z^2+x)]的值.
来源：学生作业帮助网 编辑：作业帮 时间：2024/04/30 10:26:35
![设x,y,z是实数,且(x-y)^2+(y-z)^2+(z-x)^2=(x+y-2z)^2+(y+z-2x)^2+(z+x-2y)^2求[(xy+1)(yz+1)(zx+1)]/[(x^2+1)(y^2+1)(z^2+x)]的值.](/uploads/image/z/1343996-44-6.jpg?t=%E8%AE%BEx%2Cy%2Cz%E6%98%AF%E5%AE%9E%E6%95%B0%2C%E4%B8%94%28x-y%29%5E2%2B%28y-z%29%5E2%2B%28z-x%29%5E2%3D%28x%2By-2z%29%5E2%2B%28y%2Bz-2x%29%5E2%2B%28z%2Bx-2y%29%5E2%E6%B1%82%5B%28xy%2B1%29%28yz%2B1%29%28zx%2B1%29%5D%2F%5B%28x%5E2%2B1%29%28y%5E2%2B1%29%28z%5E2%2Bx%29%5D%E7%9A%84%E5%80%BC.)
设x,y,z是实数,且(x-y)^2+(y-z)^2+(z-x)^2=(x+y-2z)^2+(y+z-2x)^2+(z+x-2y)^2求[(xy+1)(yz+1)(zx+1)]/[(x^2+1)(y^2+1)(z^2+x)]的值.
设x,y,z是实数,且(x-y)^2+(y-z)^2+(z-x)^2=(x+y-2z)^2+(y+z-2x)^2+(z+x-2y)^2求[(xy+1)(yz+1)(zx+1)]/
[(x^2+1)(y^2+1)(z^2+x)]的值.
设x,y,z是实数,且(x-y)^2+(y-z)^2+(z-x)^2=(x+y-2z)^2+(y+z-2x)^2+(z+x-2y)^2求[(xy+1)(yz+1)(zx+1)]/[(x^2+1)(y^2+1)(z^2+x)]的值.
(x+y-2z)^2+(y+z-2x)^2+(z+x-2y)^2
=((x-z)+(y-z))²+((y-x)+(z-x))²+((z-y)+(x-y))²
=(x-z)²+(y-z)²+2(x-z)(y-z)+(y-x)²+(z-x)²+2(y-x)(z-x)+(z-y)²+(x-y)²+2(z-y)(x-y)
=2(z-x)²+2(y-z)²+2(x-y)²+2(x-z)(y-z)+2(y-x)(z-x)+2(z-y)(x-y)
=(x-y)^2+(y-z)^2+(z-x)^2
所以
-((x-y)^2+(y-z)^2+(z-x)^2)=2(x-z)(y-z)+2(y-x)(z-x)+2(z-y)(x-y)
-2x²-2y²+2xy-2z²+2yz+2zx=2xy-2xz-2zy+2z²+2yz-2xy-2xz+2x²+2xz-2zy-2xy+2y²
-2x²-2y²+2xy-2z²+2yz+2zx=-2xz-2zy+2z²-2xy+2x²+2y²
4x²+4y²+4z²=4xy+4zy+4xz
即x²+y²+z²-xy-zy-xz=0
(1/2)(x²+y²-2xy+x²+z²-2xz+y²+z²-2zy)=0
(1/2)((x-y)²+(x-z)²+(y-z)²)=0
所以x=y=z
[(xy+1)(yz+1)(zx+1)]/ [(x^2+1)(y^2+1)(z^2+1)]
=(x²+1)(y²+1)(z²+1)/[(x^2+1)(y^2+1)(z^2+1)]
=1