Question

[Yên Lê Thanh]

Cho \(\frac{a+b}{2007}=\frac{b+c}{2008}=\frac{c+a}{2009}\)

CMR: 4(a-c)(b-a)=(c-b)²

Answer 1

[Mộc Khả Di]

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)