Đáp án:
$\begin{array}{l}
Đặt:\frac{x}{y} = \frac{y}{z} = t\\
\Rightarrow \left\{ \begin{array}{l}
x = y.t\\
y = z.t
\end{array} \right.\\
\Rightarrow x = z.t.t = z.{t^2}\\
\Rightarrow \frac{x}{z} = {t^2}\\
\frac{{{x^2} + {y^2}}}{{{y^2} + {z^2}}} = \frac{{{{\left( {y.t} \right)}^2} + {y^2}}}{{{{\left( {z.t} \right)}^2} + {z^2}}} = \frac{{{y^2}.\left( {{t^2} + 1} \right)}}{{{z^2}\left( {{t^2} + 1} \right)}} = \frac{{{y^2}}}{{{z^2}}} = {t^2}\\
\Rightarrow \frac{{{x^2} + {y^2}}}{{{y^2} + {z^2}}} = \frac{x}{z}
\end{array}$
