\[\begin{array}{l} \frac{{a + b - c}}{c} = \frac{{b + c - a}}{a} = \frac{{c + a - b}}{b} = \frac{{a + b - c + b + c - a + c + a - b}}{{c + a + b}} = \frac{{a + b + c}}{{c + a + b}} = 1\\ \to a + b = 2c;b + c = 2a;c + a = 2b \to a = b = c\\ \to (1 + \frac{b}{a})(1 + \frac{c}{b})(1 + \frac{a}{c}) = 2.2.2 = 8 \end{array}\]
