Given a Triangle, Find Something Related To Trigonometry

This can be easily attacked via the sum of three Law-of-Cosine terms. Just utilize the three given side lengths, which one ends up with $\boxed{\frac{97}{65}}$ as the result.

Recall the relation $\cos A + \cos B + \cos C = 1 + \frac{r}{R}$ .

You know that the area is $84$ through right triangles or heron's formula.

Thus you can find $r, R$ through $rs = K$ $R = \frac{abc}{4K}$

Simplifying, you get that $r = 4$ and $R = \frac{65}{8}$ Therefore, the expression is $\frac{97}{65} \implies 97 + 65 = \boxed{162}$