Đáp án: Đặt
$\begin{array}{l}
\frac{a}{b} = \frac{b}{c} = \frac{c}{a} = k \Rightarrow \left\{ \begin{array}{l}
a = b.k\\
b = c.k\\
c = a.k
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
a = \left( {c.k} \right).k = c.{k^2}\\
b = c.k
\end{array} \right.\\
\Rightarrow M = \frac{{\left( {{a^2} + {b^2} + {c^2}} \right)}}{{{{\left( {a + b + c} \right)}^2}}}\\
= \frac{{{{\left( {c.{k^2}} \right)}^2} + {c^2}.{k^2} + {c^2}}}{{{{\left( {c.{k^2} + c.k + c} \right)}^2}}}\\
= \frac{{{c^2}.{k^4} + {c^2}.{k^2} + {c^2}}}{{{c^2}.{{\left( {{k^2} + k + 1} \right)}^2}}}\\
= \frac{{{c^2}\left( {{k^4} + {k^2} + 1} \right)}}{{{c^2}{{\left( {{k^2} + k + 1} \right)}^2}}}\\
= \frac{{{k^4} + {k^2} + 1}}{{{{\left( {{k^2} + k + 1} \right)}^2}}}
\end{array}$
