If `a+b+c= 0`, then
`(a+b)/(c ) + (b+c)/(a) + (c +a)/(b)`= ?

Answer

To solve the expression \(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\) given that \(a + b + c = 0\), we can follow these steps:

### Step 1: Express \(a + b\), \(b + c\), and \(c + a\) in terms of \(c\), \(a\), and \(b\) respectively.
Since \(a + b + c = 0\), we can rearrange this equation to express \(a + b\), \(b + c\), and \(c + a\) as follows:
- \(a + b = -c\)
- \(b + c = -a\)
- \(c + a = -b\)

### Step 2: Substitute these expressions into the original equation.
Now, we substitute these values into the expression:
\[ \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} = \frac{-c}{c} + \frac{-a}{a} + \frac{-b}{b} \]

### Step 3: Simplify each term.
Now we simplify each term:
- \(\frac{-c}{c} = -1\)
- \(\frac{-a}{a} = -1\)
- \(\frac{-b}{b} = -1\)

### Step 4: Combine the simplified terms.
Now we combine these simplified terms:
\[ -1 - 1 - 1 = -3 \]

### Final Answer:
Thus, the value of \(\frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b}\) is \(-3\).

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