\(\begin{array}{l} \frac{a}{{b + c}} = \frac{b}{{a + c}} = \frac{c}{{a + b}} = \frac{{a + b + c}}{{b + c + a + c + a + b}} = \frac{1}{2}\\ \Rightarrow \left\{ \begin{array}{l} b + c = 2a\\ a + c = 2b\\ a + b = 2c \end{array} \right.\\ \Rightarrow a = b = c\\ M = \frac{{a + b}}{b}.\frac{{b + c}}{c}.\frac{{c + a}}{a}\\ M = 8 \end{array}\)
