Let  $PQR$  be a triangle. Let  $\overline{a}=\overline{QR},\overline{b}=\overline{RP},\overline{c}=\overline{PQ}$ . If  $\left|\overline{a}\right|=12,\left|\overline{b}\right|=4\sqrt{3}$  and  $\overline{b}.\overline{c}=24$ , then which of the following are true ______

$$\begin{gathered} \overline{a}+\overline{b}+\overline{c}=0 \Rightarrow \overline{b}+\overline{c}=-\overline{a} \\ {\left|\overline{b}\right|}^2+{\left|\overline{c}\right|}^2+2(\overline{b}.\overline{c})={\left|\overline{a}\right|}^2 \Rightarrow \left|\overline{c}\right|=4\sqrt{3} \\ \frac{{\left|\overline{c}\right|}^2}{2}-\left|\overline{a}\right|=24-12=12 \Rightarrow \frac{{\left|\overline{c}\right|}^2}{2}+\left|\overline{a}\right|=36 \\ \overline{a}+\overline{b}=-\overline{c} \Rightarrow {\left|\overline{a}\right|}^2+{\left|\overline{b}\right|}^2+2(\overline{a}.\overline{b})={\left|\overline{c}\right|}^2 \Rightarrow \overline{a}.\overline{b}=-72 \end{gathered}$$