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Using ratio/proportion properties,

\(\frac{{b + c}}{{11}} = \frac{{c + a}}{{12}} = \frac{{a + b}}{{13}} = \frac{{2(a + b + c)}}{{11 + 12 + 13}}\\ \Rightarrow \frac{{b + c}}{{11}} = \frac{{c + a}}{{12}} = \frac{{a + b}}{{13}} = \frac{{a + b + c}}{{18}} \\ \Rightarrow \frac{{b + c}}{{11}} = \frac{{c + a}}{{12}} = \frac{{a + b}}{{13}} = \frac{{a + b + c}}{{18}} = \frac{a}{7} = \frac{b}{6} = \frac{c}{5}\)

Let a=7k, b=6k, c=5k

\(cos A=\frac {1}{5}, cos B=\frac {19}{35}, cos C=\frac {5}{7}\)

Rest should take a few more steps.