Assume $ \vec{r}' \perp d\vec{l} $, and $ \vec{B} = \mathrm{const} $ (constant vector field).
How to prove the identity: \begin{equation*} \oint\limits_L \vec{r}'\times( d\vec{l} \times \vec{B}) = - \vec{B} \times \frac12 \oint\limits_L \left( \vec{r}'\times d\vec{l}\right) \end{equation*} where $L$ is a closed loop.
How to take out a constant vector from this kind of integral? (I don't understand where the factor 1/2 comes from, but the formula is right).