Question

已知：(a-b)(b-c)(c-a)/(a+b)(b+c)(c+a)=5/132,求a/(a+b)+b/(b+c)+c/(c+a)的值

Answer 1

设a+b=2x (1)
b+c=2y (2)
c+a=2z (3)
则：a+b+c=x+y+z (4)
(1)-(2):a-c=2(x-y) (5)
(2)-(3):b-a=2(y-z) (6)
(3)-(1):c-b=2(z-x) (7)
(4)-(2):a=x+z-y (8)
(4)-(3):b=y+x-z (9)
(4)-(1):c=y+z-x (10)
则：(a-b)(b-c)(c-a)/(a+b)(b+c)(c+a)
=2(z-y)2(x-z)2(y-x)/2x2y2z
=(zxy-zxx-zzy+zzx-yxy+yxx+yzy-yzx)/xyz
=-x/y-z/x+z/y-y/z+x/z+y/x=5/132
则：a/(a+b)+b/(b+c)+c/(c+a)
=(x+z-y)/x+(y+x-z)/y+(y+z-x)/z
=1+z/x-y/x+1+x/y-z/y+y/z+1-x/z
=3-5/132=391/132