Từ : \(\frac{a}{b}=\frac{b}{c}\Rightarrow\frac{a}{b}.\frac{a}{b}=\frac{b}{c}.\frac{b}{c}\Rightarrow\frac{a^2}{b^2}=\frac{b^2}{c^2}=\frac{a^2+b^2}{c^2+b^2}\)
Từ : \(\frac{a}{b}=\frac{b}{c}\Rightarrow a.c=b^2\) .
Mà : \(\frac{a^2+b^2}{b^2+c^2}=\frac{b^2}{c^2}\) ; thay \(b^2=a.c\) ta có :
\(\frac{a^2+b^2}{b^2+c^2}=\frac{a.c}{c^2}=\frac{a}{c}\) (đpcm)
\(\frac{a}{b}\) = \(\frac{b}{c}\) thì \(\frac{a^2+b^2}{b^2+c^2}\) =\(\frac{a}{b}\)
Mà \(\frac{a}{b}\)= \(\frac{b}{c}\) => a.c = b.b
Ta có \(\frac{a^2+b^2}{b^2+c^2}\) = \(\frac{a.a+b.b}{b.b+c.c}\) = \(\frac{a.a+a.c}{a.c+c.c}\) = \(\frac{a.\left(a+c\right)}{c.\left(a+c\right)}\)= \(\frac{a}{c}\)
