Prove some properties of the scalar triple product

Use the properties of the cross product and the dot product to prove the following properties of the scalar triple product.

Proof. We have
$(A+B) \cdot (A+B) \times C = (A+B) \cdot ((A+B) \times C) = 0$

since $D \cdot (D \times E) = 0$ for any vectors $D,E$ (in this case $D = A+B$ and $E = C). \qquad blacksquare$
Proof. Using part (a), we have
$\begin{align*} (A+B) \cdot ((A+B) \times C) = 0 && \implies && (A+B) \cdot ((A \times C) + (B \times C)) &= 0 \\ && \implies && B \cdot (A \times C) + A \cdot (B \times C) &= 0 \\ && \implies && A \cdot (B \times C) &= -B \cdot (A \times C). \qquad \blacksquare \end{align*}$
Proof. We have
$A \cdot B \times C = A \cdot (-C \times B) = A \cdot (-1)(C \times B) = -A \cdot (C \times B). \qquad \blacksquare$
Proof. We have
$\begin{align*} A \cdot B \times C &= -A \cdot C \times B &(\text{part c}) \\ &= C \cdot A \times B &(\text{part b}) \\ &= -C \cdot B \times A &(\text{part c}). \qquad \blacksquare \end{align*}$