The vector triple product reads: $$ \vec{a} \times(\vec{b}\times \vec{c})=(\vec a\cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} \tag{1} $$
In proving $(1)$ geometrically, I understand that the resulting vector must lie on the plane spanned by $\vec{b}$ and $\vec{c}$, and also orthogonal to $\vec{a}$, hence: $$ \vec{a} \times(\vec{b}\times \vec{c})=m(\vec a\cdot \vec{c})\vec{b} - m(\vec{a} \cdot \vec{b})\vec{c} $$ However, I am having trouble proving the magnitude of the resulting vector.
My question is: Why is $m=1$?
Please note that I am looking for a geometric proof.