Đáp án:
$\begin{array}{l}
\dfrac{a}{b} = \dfrac{b}{c}\\
\Rightarrow a.c = b.b\\
\Rightarrow {b^2} = a.c\\
Thay\,{b^2} = a.c\,\text{vào}:\\
\dfrac{{{a^2} + {b^2}}}{{{b^2} + {c^2}}} = \dfrac{{{a^2} + a.c}}{{a.c + {c^2}}} = \dfrac{{a.\left( {a + c} \right)}}{{c.\left( {a + c} \right)}} = \dfrac{a}{c}\\
\text{Vậy}\,\dfrac{{{a^2} + {b^2}}}{{{b^2} + {c^2}}} = \dfrac{a}{c}
\end{array}$