A marriage between geometry and algebra

Note that $\frac{b+c}{\sin A}+\frac{a+c}{\sin B}+\frac{a+b}{\sin C}=(a+b+c)\left(\frac{1}{\sin A}+\frac{1}{\sin B}+\frac{1}{\sin C}\right)-6R$ from extended sine rule . Furthermore, by Cauchy-Schwarz inequality , we have $(a+b+c)\left(\frac{1}{\sin A}+\frac{1}{\sin B}+\frac{1}{\sin C}\right)\geq\left(\sqrt{\frac{a}{\sin A}}+\sqrt{\frac{b}{\sin B}}+\sqrt{\frac{c}{\sin C}}\right)=(3\sqrt{2R})^2=18R.$ Hence, $K=18-6=12.$ Equality holds when $\Delta ABC$ is an equilateral triangle.

From the sine rule: $\frac{a}{sin{A}} = \frac{b}{sin{B}} = \frac{c}{sin{C}}\ = 2R$

Thus, for an angle X and corresponding side x: $sin{X} = \frac{x}{2R}$

Using this in the LHS: $\frac{b+c}{\sin A}+\frac{a+c}{\sin B}+\frac{a+b}{\sin C} = 2R * (\frac{b+c}{a} + \frac{c+a}{b} + \frac{a+b}{c})$

From the AM-GM inequality (since $a,b,c > 0$ ) : $\frac{\frac{b+c}{a} + \frac{c+a}{b} + \frac{a+b}{c}}{6} \geq 1$ $=> \frac{b+c}{a} + \frac{c+a}{b} + \frac{a+b}{c} \geq 6$ The equality holds when $a=b=c$ .

Thus, $K = 12$ .