Let $a,b,c,x$ be vectors in $\mathbb{R}^3$, such that $a,b,c$ are linearly independent. Then the following equation holds: $$(x\cdot a)(b\times c)+(x\cdot b)(c\times a)+(x\cdot c)(a\times b)=x[a,b,c],$$ where $[a,b,c]:=(a\times b)\cdot c$.
I came up with the following proof:
Denote $y=(x\cdot a)(b\times c)+(x\cdot b)(c\times a)+(x\cdot c)(a\times b)-x[a,b,c]$. Then it's easy to check by the properties of the mixed product that $y\cdot a=y\cdot b=y\cdot c=0$. Since $a,b,c$ are L.I., $y$ must be zero, thus verifying the equation.
However, I feel this proof a bit lucky, and I wonder whether the equation has any intuitive or geometrical interpretation, or anything that makes the equation more intelligible? Any idea is welcomed.