To solve the problem where (\frac{a+b}{c} = \frac{b+c}{a} = \frac{c+a} = k), we proceed as follows:
From the given ratios:
[a + b = ck \quad \text{(1)}]
[b + c = ak \quad \text{(2)}]
[c + a = bk \quad \text{(3)}]
Adding (1), (2), and (3):
[2(a + b + c) = k(a + b + c)]
Case 1: (a + b + c \neq 0)
We can divide both sides by (a + b + c):
[k = 2]
Case 2: (a + b + c = 0)
If (a + b + c = 0), then (a + b = -c). Substitute into (\frac{a+b}{c}):
[k = \frac{-c}{c} = -1]
The possible values of (k) are (\boxed{2}) and (\boxed{-1}).
Answer: (\boxed{2}) or (\boxed{-1}) (both are valid solutions)
If the problem expects both answers, write them as (\boxed{2}) and (\boxed{-1}). For most standard contexts, these are the two correct values.
(\boxed{2}) and (\boxed{-1})
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