Let $\overrightarrow{\mathrm{a}},\overrightarrow{\mathrm{b}},\overrightarrow{\mathrm{c}}$ be three vectors representing sides of a triangle such that $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0}$ and angle between $\overrightarrow{\mathrm{a}}\text{ and }\overrightarrow{\mathrm{b}}$ is $\frac{\mathrm{\pi }}{3}$. If $\left|\overrightarrow{\mathrm{a}}\right|=\left|\overrightarrow{\mathrm{b}}\right|\text{ and }\left|\overrightarrow{\mathrm{c}}\right|=6$, then:

$\begin{array}{l}\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=0\\ \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=-\overrightarrow{\mathrm{c}}\\ {\mathrm{a}}^2+{\mathrm{b}}^2+2\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}={\mathrm{c}}^2\\ {\mathrm{a}}^2+{\mathrm{b}}^2-{\mathrm{c}}^2=-\mathrm{ab}\\ \frac{{\mathrm{a}}^2+{\mathrm{b}}^2-{\mathrm{c}}^2}{2\mathrm{ab}}=\frac{-1}{2}\end{array}$

$\begin{array}{l}\Rightarrow \mathrm{Cos}\mathrm{C}=\frac{-1}{2}=\mathrm{\angle }\mathrm{C}=\frac{2\mathrm{\pi }}{3}\\ \left|\overrightarrow{\mathrm{a}}\right|+\left|\overrightarrow{\mathrm{b}}\right|=2\sqrt{3}\end{array}$
$\therefore$Area of the triangle $=\frac{1}{2}\times 2\sqrt{3}\times 2\sqrt{3}\times \frac{\sqrt{3}}{2}=3\sqrt{3}$ nd length of the median
bisecting the vector $\overrightarrow{\mathrm{c}},\mathrm{\ell }=3\tan {30}^{\circ }=\sqrt{3}$