If `a^(2)+ab+b^(2) = b^(2) +bc +c^(2)` where `a ne b ne c` then find the value of `a+b+c`.

Answer

To solve the equation \( a^2 + ab + b^2 = b^2 + bc + c^2 \) under the condition that \( a \neq b \neq c \), we can follow these steps:

### Step 1: Simplify the equation
Start with the given equation:
\[ a^2 + ab + b^2 = b^2 + bc + c^2 \]
Subtract \( b^2 \) from both sides:
\[ a^2 + ab = bc + c^2 \]

**Hint:** Look for terms that can be canceled out or moved to one side of the equation.

### Step 2: Rearrange the equation
Rearranging gives:
\[ a^2 + ab - c^2 - bc = 0 \]
This can be rewritten as:
\[ a^2 + ab - c^2 - bc = 0 \]

**Hint:** Group similar terms together to see if you can factor or simplify further.

### Step 3: Factor the equation
Notice that \( a^2 - c^2 \) can be factored using the difference of squares:
\[ (a - c)(a + c) + b(a - c) = 0 \]
This can be factored as:
\[ (a - c)(a + c + b) = 0 \]

**Hint:** Look for common factors in the equation.

### Step 4: Solve for \( a - c \)
Since \( a \neq c \) (given), we can conclude:
\[ a + c + b = 0 \]

**Hint:** If one factor is zero and it doesn't apply due to the conditions, set the other factor to zero.

### Step 5: Find \( a + b + c \)
From the equation \( a + c + b = 0 \), we can rearrange it to find:
\[ a + b + c = 0 \]

**Hint:** Rearranging the equation can often lead you directly to the answer.

### Final Answer
Thus, the value of \( a + b + c \) is:
\[ \boxed{0} \]