已知 (x+y-z)/z=(x-y+z)/y=(y+z-x)/x,且xyz≠0,求代数式 ((x+y)(y+z)(x+z))/xyz

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已知 (x+y-z)/z=(x-y+z)/y=(y+z-x)/x,且xyz≠0,求代数式 ((x+y)(y+z)(x+z))/xyz
已知 (x+y-z)/z=(x-y+z)/y=(y+z-x)/x,且xyz≠0,求代数式 ((x+y)(y+z)(x+z))/xyz

已知 (x+y-z)/z=(x-y+z)/y=(y+z-x)/x,且xyz≠0,求代数式 ((x+y)(y+z)(x+z))/xyz
设x+y-z/z=x-y+z/y=y+z-x/x=k
有x+y-z=kz
x-y+z=ky
y+z-x=kx
三式相加得x+y+z=k（x+y+z） k=1
得x+y=（k+1）z
x+z=（k+1）y
y+z=（k+1）x
当x+y+z≠0时
(x+y)(y+z)(x+z)/xyz=（k+1）^3xyz/xyz=(k+1)^3=8
当x+y+z=0时
x+y=-z,x+z=-y,y+z=-x
代入得-1
所以是-1或8

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