Example 9 If $\frac{a+b}{c}=\frac{b+c}{a}=\frac{c+a}{b}=k$, then the graph of the line $y=k x+k$ must pass through ().
(A) the first, second, and third quadrants
(B) the first and third quadrants
(C) the second, third, and fourth quadrants
(D) none of the above is correct
(1995, Hebei Province Junior High School Mathematics Competition)
Solution: When $a+b+c \neq 0$, $\frac{2(a+b+c)}{a+b+c}=k$, $k=2$.

When $a+b+c=0$, $a+b=-c, b+c=-a, c+a=-b$, we get $k=-1$.

Therefore, when $a+b+c \neq 0$, the line $y=kx+k$ is such that when $a+b+c=0$, the line $y=kx+k$ becomes $y=-x-1$, at this time the graph passes through the second, third, and fourth quadrants.

In summary, the graph must pass through the second and third quadrants, so the answer is (B).