\(\dfrac{a}{c}=\dfrac{c}{b}\\ \Rightarrow ab=c^2\\ \Rightarrow\dfrac{a^2+c^2}{b^2+c^2}=\dfrac{a^2+ab}{b^2+ab}\\ =\dfrac{a\left(a+b\right)}{b\left(a+b\right)}\\ =\dfrac{a}{b}\)
Đặt \(\dfrac{a}{c}=\dfrac{c}{b}=k\)
\(\Rightarrow a=c.k;c=b.k\)
\(\Rightarrow\dfrac{a^2+c^2}{b^2+c^2}=\dfrac{\left(c.k\right)^2+\left(b.k\right)^2}{b^2+\left(b.k\right)^2}=\dfrac{k^2.\left(c^2+b^2\right)}{b^2.\left(k^2+1\right)}\)
\(=\dfrac{k^2.\left[\left(b.k^2\right)+b^2\right]}{b^2.\left(k^2+1\right)}=\dfrac{k^2.\left[b^2.\left(k^2+1\right)\right]}{b^2.\left(k^2+1\right)}=k^2\left(1\right)\)
\(\dfrac{a}{b}=\dfrac{c.k}{b}=\dfrac{b.k^2}{b}=k^2\left(2\right)\)
Từ ( 1 ) và ( 2 ) \(\Rightarrow\dfrac{a^2+c^2}{b^2+c^2}=\dfrac{a}{b}\left(đpcm\right)\)
