If a+b+c= 0, then the value of `((a+b)/c + (b+c)/a + (c+a)/b)`
`((a)/(b+c) + b/(c+a) + c/(a+b))` is :

Options

A. 8
B. `-3`
C. 9
D. 0

Answer

To solve the problem, we start with the given condition \( a + b + c = 0 \). We need to find the value of the expression:

\[ \frac{(a+b)}{c} + \frac{(b+c)}{a} + \frac{(c+a)}{b} \quad \text{and} \quad \frac{a}{(b+c)} + \frac{b}{(c+a)} + \frac{c}{(a+b)} \]

### Step 1: Simplify the first expression

Using the condition \( a + b + c = 0 \), we can express \( c \), \( a \), and \( b \) in terms of the other variables:

- \( c = - (a + b) \)
- \( a = - (b + c) \)
- \( b = - (c + a) \)

Now, substituting these into the first expression:

\[ \frac{(a+b)}{c} = \frac{(a+b)}{- (a+b)} = -1 \]

\[ \frac{(b+c)}{a} = \frac{(b - (a+b))}{a} = \frac{-a}{a} = -1 \]

\[ \frac{(c+a)}{b} = \frac{(- (a+b) + a)}{b} = \frac{-b}{b} = -1 \]

Adding these results together:

\[ -1 - 1 - 1 = -3 \]

### Step 2: Simplify the second expression

Now, we simplify the second expression:

\[ \frac{a}{(b+c)} = \frac{a}{-a} = -1 \]

\[ \frac{b}{(c+a)} = \frac{b}{-b} = -1 \]

\[ \frac{c}{(a+b)} = \frac{- (a+b)}{-(a+b)} = 1 \]

Adding these results together:

\[ -1 - 1 + 1 = -1 \]

### Step 3: Compare the two results

Now we compare the two simplified expressions:

1. The first expression evaluates to \(-3\).
2. The second expression evaluates to \(-1\).

Thus, we have:

\[ -3 < -1 \]

### Conclusion

Therefore, the value of the expression is:

\[ \frac{(a+b)}{c} + \frac{(b+c)}{a} + \frac{(c+a)}{b} < \frac{a}{(b+c)} + \frac{b}{(c+a)} + \frac{c}{(a+b)} \]

### Final Answer

The final answer is:

\[ \text{The value is } -3 < -1. \]