I am trying to prove that \begin{equation*} (\vec{a} \times \vec{b}) \times (\vec{a} \times \vec{c}) = (\vec{a} \cdot (\vec{b} \times \vec{c}))\vec{a}. \end{equation*} I'd have a lot of false starts with this problem. I tried swapping the dot and cross product without any luck. I tried working from either the left-hand or the right-hand side, but neither seemed particularly easier.
I would really appreciate some direction on this. A hint while I continue to work on this would be greatly appreciated.
Set $d=a\times c$. Then $$(a\times b)\times d=(a\cdot d)b-(b\cdot d)a$$ by the vector triple product identity. But $a$ and $d$ are orthogonal and also $$b\cdot d=b\cdot(a\times c)$$ a scalar triple product etc.