If $\overrightarrow{\mathrm{a}}\times (\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}})$ is perpendicular to $(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}}$ we may have

$\begin{array}{r}\overrightarrow{\mathrm{a}}\times (\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}})=(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}})\overrightarrow{\mathrm{b}}-(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}})\overrightarrow{\mathrm{c}}\\ \mathrm{and} (\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}}=-(\overrightarrow{\mathrm{c}}\cdot \overrightarrow{\mathrm{b}})\overrightarrow{\mathrm{a}}+(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}})\overrightarrow{\mathrm{b}}\end{array}$

We have been given
$\begin{array}{l}(\overrightarrow{\mathrm{a}}\times (\overrightarrow{\mathrm{b}}\times \overrightarrow{\mathrm{c}}\left)\right)\cdot \left(\right(\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times \overrightarrow{\mathrm{c}})=0\\ \therefore \left(\right(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}})\overrightarrow{\mathrm{b}}-(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}\left)\overrightarrow{\mathrm{c}}\right)\cdot \left(\right(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}})\overrightarrow{\mathrm{b}}-(\overrightarrow{\mathrm{c}}\cdot \overrightarrow{\mathrm{b}}\left)\overrightarrow{\mathrm{a}}\right)=0\\ \text{ or }(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}}{)}^2|\overrightarrow{\mathrm{b}}{|}^2-(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}})(\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}})(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}})\\ -(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}})(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}})(\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}})+(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}})(\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}})(\overrightarrow{\mathrm{c}}\cdot \overrightarrow{\mathrm{a}})=0\\ \text{ or }(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}}{)}^2|\overrightarrow{\mathrm{b}}{|}^2=(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}})(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}})(\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}})\\ \text{ or }(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}})\left(\right(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}}\left)\right(\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{b}})-(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}\left)\right(\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}}\left)\right)=0\\ \overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}}=0\text{ or }(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{c}})|\overrightarrow{\mathrm{b}}{|}^2=(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}}\left)\right(\overrightarrow{\mathrm{b}}\cdot \overrightarrow{\mathrm{c}})\end{array}$