计算：（x-y）(z-x)／(x-2y+z)(x+y-2z)+(z-y)(x-y)／(x+y-2z)(y+z-2x)+(x-z)(y-z)／(y+z-2x)(x-2y+z)

Question

计算：（x-y）(z-x)／(x-2y+z)(x+y-2z)+(z-y)(x-y)／(x+y-2z)(y+z-2x)+(x-z)(y-z)／(y+z-2x)(x-2y+z)

Answer 1

令x-y=a x-z=b y-z=c
（x-y）(z-x)／(x-2y+z)(x+y-2z)+(z-y)(x-y)／(x+y-2z)(y+z-2x)+(x-z)(y-z)／(y+z-2x)(x-2y+z)
=-ab/(a-c)(b+c)-ac/(b+c)(-a-b)+bc/(-a-b)(a-c)
=[-ab(-a-b)-ac(a-c)+bc(b+c)]/(a-c)(b+c)(-a-b)
=(aab+abb-aac+acc+bbc+bcc)/(ab+ac-bc+cc)(-a-b)
=(aab+abb-aac+acc+bbc+bcc)/(-aab-aac+abc-acc-abb-abc+bbc-bcc)
=(aab+abb-aac+acc+bbc+bcc)/-(aab+aac+acc+abb-bbc+bcc)
=-1