9. (16 points) Define the cross product of vectors
$$
|\boldsymbol{a} \times \boldsymbol{b}|=|\boldsymbol{a}||\boldsymbol{b}| \sin \langle\boldsymbol{a}, \boldsymbol{b}\rangle,
$$

where $\langle\boldsymbol{a}, \boldsymbol{b}\rangle$ denotes the angle between vectors $\boldsymbol{a}$ and $\boldsymbol{b}$. If three vectors $\boldsymbol{a}$, $\boldsymbol{b}$, and $\boldsymbol{c}$ satisfy $\boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=\mathbf{0}$, and their magnitudes are equal, find
$$
\left|\frac{|a \times b+b \times c+c \times a|}{a \cdot b+b \cdot c+c \cdot a}\right|
$$

the range of values.
Lemma $1 \quad a \times b=-b \times a, a \times a=0$.
Proof of Lemma 1: This is evident from the definition.
Lemma 2 If $\boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=\mathbf{0}$, then
$$
a \times b=b \times c=c \times a .
$$

Proof of Lemma 2: Note that,
$$
\begin{array}{l}
(a+b+c) \times a=a \times a+b \times a+c \times a=0 \\
\Rightarrow a \times b=b \times c .
\end{array}
$$

Similarly, $a \times b=b \times c=c \times a$.
Let $|a|=|b|=|c|=r$.
$$
\begin{array}{l}
\text { By } \boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=\mathbf{0} \\
\Rightarrow|\boldsymbol{a}|^{2}+|\boldsymbol{b}|^{2}+|\boldsymbol{c}|^{2}+2(\boldsymbol{a} \cdot \boldsymbol{b}+\boldsymbol{b} \cdot \boldsymbol{c}+\boldsymbol{a} \cdot \boldsymbol{c})=0 \\
\Rightarrow \boldsymbol{a} \cdot \boldsymbol{b}+\boldsymbol{b} \cdot \boldsymbol{c}+\boldsymbol{c} \cdot \boldsymbol{a}=-\frac{3}{2} r^{2} . \\
\text { Hence }\left|\frac{|\boldsymbol{a} \times \boldsymbol{b}+\boldsymbol{b} \times \boldsymbol{c}+\boldsymbol{c} \times \boldsymbol{a}|}{\boldsymbol{a} \cdot \boldsymbol{b}+\boldsymbol{b} \cdot \boldsymbol{c}+\boldsymbol{c} \cdot \boldsymbol{a}}\right| \\
=\frac{3|\boldsymbol{a}||\boldsymbol{b}| \sin \theta}{\frac{3}{2} r^{2}}=2 \sin \theta \in[-2,2] .
\end{array}
$$