If $\overline{\mathrm{a}},\overline{\mathrm{b}},\overline{\mathrm{c}}$ are three vectors such that $\left|\overline{\mathrm{b}}\right|=\left|\overline{\mathrm{c}}\right|$ then $\left\{(\overline{\mathrm{a}} + \overline{\mathrm{b}})\times (\overline{\mathrm{a}} + \overline{\mathrm{c}})\right\}\times \left\{(\overline{\mathrm{b}}\times \overline{\mathrm{c}})\right\}.(\overline{\mathrm{b}} + \overline{\mathrm{c}})=$

$\begin{array}{rcl}\left|\overline{\mathrm{b}}\right|& =& \left|\overline{\mathrm{c}}\right|\\ & & \left\{0+\overline{\mathrm{a}}\times \overline{\mathrm{c}}+\overline{\mathrm{b}}\times \overline{\mathrm{a}}+\overline{\mathrm{b}}\times \overline{\mathrm{c}}\right\}\times \left\{\overline{\mathrm{b}}\times \overline{\mathrm{c}}\right\}.(\overline{\mathrm{b}}+\overline{\mathrm{c}})\\ & & \left(\left(\overline{\mathrm{a}}\times \overline{\mathrm{c}}\right)\times \left(\overline{\mathrm{b}}\times \overline{\mathrm{c}}\right)+\left(\overline{\mathrm{b}}\times \overline{\mathrm{a}}\right)\times \left(\overline{\mathrm{b}}\times \overline{\mathrm{c}}\right)\right).\left(\overline{\mathrm{b}}+\overline{\mathrm{c}}\right)\\ & & \left(0-\left[\overline{\mathrm{a}} \overline{\mathrm{c}} \overline{\mathrm{b}}\right]\overline{\mathrm{c}}+\left[\overline{\mathrm{b}} \overline{\mathrm{a}} \overline{\mathrm{c}}\right]\overline{\mathrm{b}}-0\right).\left(\overline{\mathrm{b}}+\overline{\mathrm{c}}\right)\\ & & \left[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}\right]\left(\overline{\mathrm{b}}-\overline{\mathrm{c}}\right).\left(\overline{\mathrm{b}}+\overline{\mathrm{c}}\right)\\ & =& \left[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}\right]\left({\left|\overline{\mathrm{b}}\right|}^2-{\left|\overline{\mathrm{c}}\right|}^2\right)\\ & =& 0\end{array}$