If the four points $\overline{\mathrm{a}},\overline{\mathrm{b}},\overline{\mathrm{c}},\overline{\mathrm{d}}$ are coplanar then $\left[\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{c}}\right]+\left[\overline{\mathrm{c}} \overline{\mathrm{a}} \overline{\mathrm{d}}\right]+\left[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{d}}\right]=$

$\overline{\mathrm{a}},\overline{\mathrm{b}},\overline{\mathrm{c}},\overline{\mathrm{d}}$  are coplanar

$\begin{array}{rcl} & \Rightarrow & \left[\overline{\mathrm{AB}} \overline{\mathrm{AC}} \overline{\mathrm{AD}}\right]=0\\ (\overline{\mathrm{b}}-\overline{\mathrm{a}})·\left(\right(\overline{\mathrm{c}}-\overline{\mathrm{a}})\times (\overline{\mathrm{d}}-\overline{\mathrm{a}}\left)\right)& =& 0\\ (\overline{\mathrm{b}}-\overline{\mathrm{a}})·(\overline{\mathrm{c}}\times \overline{\mathrm{d}}-\overline{\mathrm{c}}\times \overline{\mathrm{a}}-\overline{\mathrm{a}}\times \overline{\mathrm{d}}+\overline{0})& =& 0\\ [\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{d}}]-[\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{a}}]-[\overline{\mathrm{b}} \overline{\mathrm{a}} \overline{\mathrm{d}}]-[\overline{\mathrm{a}} \overline{\mathrm{c}} \overline{\mathrm{d}}]& =& 0\\ [\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{d}}]+[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{d}}]+[\overline{\mathrm{c}} \overline{\mathrm{a}} \overline{\mathrm{d}}]& =& [\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]\end{array}$