Đặt :
\(\frac{a}{b}=\frac{c}{d}=k\)
\(\Rightarrow\begin{cases}a=bk\\b=ck\end{cases}\)\(\Rightarrow\begin{cases}a=ck^2\\b=ck\end{cases}\)
Thay vào ta có :
\(\frac{a^2+2b^2}{b^2+2c^2}=\frac{c^2k^4+4c^2k^2}{c^2k^2+4c^2}=\frac{c^2k^2\left(k^2+4\right)}{c^2\left(k^2+4\right)}=k^2=\frac{a}{b}.\frac{b}{c}=\frac{a}{c}\)
\(\Rightarrow\frac{a^2+2b^2}{b^2+2c^2}=\frac{a}{c}\)