x+y+z≠0.且（y+z）分之x=（z+x）分之y=（x+y）分之z.求（x+y+z）分之x

题目

x+y+z≠0.且（y+z）分之x=（z+x）分之y=（x+y）分之z.求（x+y+z）分之x

答案

令（y+z）分之x=（z+x）分之y=（x+y）分之z=k
x=k(y+z)
y=k(z+x)
z=k(x+y)
相加
x+y+z=2k(x+y+z)
(x+y+z)(2k-1)=0
x+y+z≠0
所以2k-1=0
则k=1/2
x=k(y+z)=1/2(y+z)
所以原式=[(y+z)+1/2(y+z)]分之[1/2(y+z)]
=(3/2)分之(1/2)
=3分之1