reciprocal vectors

I think they intended to write $A\cdot (B\times C)$, which is a scalar; the vectors are non-coplanar, hence this scalar is nonzero and hence we can safely divide by this scalar.

This wiki article would be a good start.

Cyclic means a cyclic permutation of the operands: $$ {\bf A} \to {\bf B}, {\bf B} \to {\bf C}, {\bf C} \to {\bf A} \quad a \to b, b \to c, c \to a $$ This gives $$ {\bf A} = \frac{{\bf b}\times {\bf c}}{{\bf a}\cdot({\bf b}\times {\bf c})} \quad {\bf B} = \frac{{\bf c}\times {\bf a}}{{\bf b}\cdot({\bf c}\times {\bf a})} \quad {\bf C} = \frac{{\bf a}\times {\bf b}}{{\bf c}\cdot({\bf a}\times {\bf b})} $$ and $$ {\bf a} = \frac{{\bf B}\times {\bf C}}{{\bf A}\cdot({\bf B}\times {\bf C})} \quad {\bf b} = \frac{{\bf C}\times {\bf A}}{{\bf B}\cdot({\bf C}\times {\bf A})} \quad {\bf c} = \frac{{\bf A}\times {\bf B}}{{\bf C}\cdot({\bf A}\times {\bf B})} $$

From Reciprocal Lattice

[..] the "crystallographer's" definition, comes from defining the reciprocal lattice to be $e^{2 \pi i\mathbf{K}\cdot\mathbf{R}}=1$ which changes the definitions of the reciprocal lattice vectors to be

$ \mathbf{b_{1}}=\frac{\mathbf{a_{2}} \times \mathbf{a_{3}}}{\mathbf{a_{1}} \cdot (\mathbf{a_{2}} \times \mathbf{a_{3}})}$ and so on for the other vectors. The crystallographer's definition has the advantage that the definition of $\mathbf{b_{1}}$ is just the reciprocal magnitude of $\mathbf{a_{1}}$ in the direction of $\mathbf{a_{2}} \times \mathbf{a_{3}}$, dropping the factor of $2 \pi$.

Checking the magnitude:

$$ \lVert{\bf A}\rVert = \frac{\lVert {\bf b} \times{\bf c}\rVert}{\lVert{\bf a}\rVert\lVert{\bf b}\times {\bf c}\rVert\cos\angle({\bf a}, {\bf b} \times{\bf c})} = \frac{1}{\lVert{\bf a}\rVert ({\bf e_a} \cdot {\bf e}_{{\bf b} \times{\bf c}})} $$

Solving the question:

Using the "bac-cab" rule ${\bf a}\times({\bf b}\times{\bf c}) = {\bf b}({\bf a}\cdot {\bf c}) - {\bf c}({\bf a}\cdot {\bf b})$ we go for the nominator of ${\bf B}\times{\bf C}$: $$ ({\bf c}\times {\bf a})\times({\bf a}\times{\bf b}) = {\bf a}(({\bf c}\times{\bf a})\cdot{\bf b})-{\bf b}(({\bf c}\times{\bf a})\cdot{\bf a})= {\bf a}(({\bf c}\times{\bf a})\cdot{\bf b}) $$ because ${\bf c}\times{\bf a} \perp {\bf a}$. This gives $$ {\bf B}\times{\bf C} = \frac{{\bf a}(({\bf c}\times{\bf a})\cdot{\bf b})}{({\bf b}\cdot({\bf c}\times {\bf a}))({\bf c}\cdot({\bf a}\times {\bf b}))} = \frac{{\bf a}}{{\bf c}\cdot({\bf a}\times {\bf b})} \\ {\bf A}\cdot({\bf B}\times{\bf C}) = \frac{{\bf b}\times {\bf c}}{{\bf a}\cdot({\bf b}\times {\bf c})} \cdot \frac{{\bf a}}{{\bf c}\cdot({\bf a}\times {\bf b})} = \frac{1}{{\bf c}\cdot({\bf a}\times {\bf b})} $$ Dividing those gives the desired result $$ \frac{{\bf B}\times{\bf C}}{{\bf A}\cdot({\bf B}\times{\bf C})} = \frac{{\bf c}\cdot({\bf a}\times {\bf b})\,{\bf a}}{{\bf c}\cdot({\bf a}\times {\bf b})} = {\bf a} $$