If ` a + b + c = 0` then `( 2 a^(2))/( b^(2) + c^(2) - a^(2)) + ( 2b^(2))/( c^(2) + a^(2) - b^(2)) + ( 2 c^(2))/( a^(2) + b^(2) - c^(2)) + 3= `

Options

A. A)3
B. B)`-4`
C. C)0
D. D)`-3`

Answer

To solve the problem, we start with the equation given:

**Given:**
\[ a + b + c = 0 \]

We need to evaluate the expression:
\[ \frac{2a^2}{b^2 + c^2 - a^2} + \frac{2b^2}{c^2 + a^2 - b^2} + \frac{2c^2}{a^2 + b^2 - c^2} + 3 \]

### Step 1: Substitute values for a, b, and c
Since \( a + b + c = 0 \), we can choose values for \( a \), \( b \), and \( c \) that satisfy this equation. Let's take:
- \( a = 1 \)
- \( b = 1 \)
- \( c = -2 \)

### Step 2: Substitute into the expression
Now, we substitute these values into the expression:

1. Calculate \( b^2 + c^2 - a^2 \):
\[ b^2 + c^2 - a^2 = 1^2 + (-2)^2 - 1^2 = 1 + 4 - 1 = 4 \]

2. Calculate \( c^2 + a^2 - b^2 \):
\[ c^2 + a^2 - b^2 = (-2)^2 + 1^2 - 1^2 = 4 + 1 - 1 = 4 \]

3. Calculate \( a^2 + b^2 - c^2 \):
\[ a^2 + b^2 - c^2 = 1^2 + 1^2 - (-2)^2 = 1 + 1 - 4 = -2 \]

### Step 3: Substitute back into the expression
Now substitute these values back into the expression:

\[ \frac{2a^2}{b^2 + c^2 - a^2} = \frac{2 \cdot 1^2}{4} = \frac{2}{4} = \frac{1}{2} \]

\[ \frac{2b^2}{c^2 + a^2 - b^2} = \frac{2 \cdot 1^2}{4} = \frac{2}{4} = \frac{1}{2} \]

\[ \frac{2c^2}{a^2 + b^2 - c^2} = \frac{2 \cdot (-2)^2}{-2} = \frac{2 \cdot 4}{-2} = \frac{8}{-2} = -4 \]

### Step 4: Combine all parts of the expression
Now, combine these results:

\[ \frac{1}{2} + \frac{1}{2} - 4 + 3 \]

Calculating this step-by-step:

1. \( \frac{1}{2} + \frac{1}{2} = 1 \)
2. \( 1 - 4 = -3 \)
3. \( -3 + 3 = 0 \)

### Final Result
Thus, the value of the expression is:
\[ 0 \]

### Conclusion
The answer is \( 0 \).