Theo tính chất dãy tỉ số bằng nhau, ta có:
\(\frac{a+b}{2007}=\frac{b+c}{2008}=\frac{a+b-\left(b+c\right)}{2007-2008}=\frac{a-c}{-1}\)(1)
\(\frac{b+c}{2008}=\frac{c+a}{2009}=\frac{b+c-\left(c+a\right)}{2008-2009}=\frac{b-a}{-1}\)(2)
\(\frac{c+a}{2009}=\frac{a+b}{2007}=\frac{c+a-\left(a+b\right)}{2009-2007}=\frac{c-b}{2}\)(3)
Từ (1), (2), (3) =>\(\frac{a-c}{-1}=\frac{b-a}{-1}=\frac{c-b}{2}\)
=> \(a-c=b-a=\frac{c-b}{2}\)
=>\(c-b=2\left(a-c\right)\)
Có: \(4\left(a-c\right)\left(b-a\right)=4\left(a-c\right)\left(a-c\right)\)
(do \(a-c=b-a\)) (*)
Có \( \left(c-b\right)^2=2\left(a-c\right).2\left(a-c\right)\)
=\(4.\left(a-c\right)\left(a-c\right)\) (**)
Từ (*) và (**) =>\(4.\left(a-c\right)\left(b-a\right)=\left(c-b\right)^2\)(đpcm)