Example 6 If $a b c \neq 0$, and $\frac{a+b}{c}=\frac{b+c}{a}=$ $\frac{c+a}{b}$, then $\frac{(a+b)(b+c)(c+a)}{a b c}=$ $\qquad$
(1985, Hangzhou City Junior High School Mathematics Competition)
Let $\frac{a+b}{c}=\frac{b+c}{a}=\frac{c+a}{b}=k$. Then $a+b=c k, b+c=a k, c+a=b k$.
We have $2(a+b+c)=k(a+b+c)$.
If $a+b+c \neq 0$, then $k=2$, the original expression $=8$;
If $a+b+c=0$, then $a+b=-c, k=$ -1. The original expression $=-1$.