Question

如果四条线段a、b、c、d满足a/b=c/d,那么（a+b)/(a-b)与(c+d)/(c-d)相等吗？为什么？

Answer 1

(a+b)/(a-b)=(a+b)/a-(a+b)/b=1+b/a-a/b-1=b/a-a/b(c+d)/(c-d)=(c+d)/c-(c+d)/d=1+d/c-c/d-1=d/c-c/d因为a/b=c/d，所以b/a=d/c所以a/b=c/d，b/a=d/c等量减等量，所以相等

Answer 2

相等证明：a/b=c/d两边同时导数 切乘以 -1-b/a = - d/c两边同时 加1(a-b)/a = (c-d)/c同时倒数a/(a-b) = c/(c-d) (1)~a/b=c/d两边同时减1(a-b)/b = (c-d)/d同时倒数b/(a-b) = d/(c-d) (2)(1) + (2) 得a+b)/(a-b) = (c+d)/(c-d)

Answer 3

相等a/b=c/d所以a/2b-1/2=c/2d-1/2所以(a-b)/2b=(c-d)/2d所以2b/(a-b)=2d/(c-d)所以1+2b/(a-b)=1+2d/(c-d)所以（a+b)/(a-b)=(c+d)/(c-d)