If `a, b, c` are sides of a triangle and `|(a^2,b^2,c^2),((a+1)^2,(b+1)^2,(c+1)^2),((a-1)^2,(b-1)^2,(c-1)^2)|=` then

Options

A. `Delta` ABC is an equilateral triangle
B. `Delta` ABC is a right angled isosceles triangle
C. `Delta` ABC is an isosceles triangle
D. None of the above

Answer

To solve the problem, we need to evaluate the determinant given by:

\[ D = \begin{vmatrix} a^2 & b^2 & c^2 \\ (a+1)^2 & (b+1)^2 & (c+1)^2 \\ (a-1)^2 & (b-1)^2 & (c-1)^2 \end{vmatrix} \]

### Step 1: Expand the second and third rows

First, we expand the elements of the second and third rows:
- The second row becomes:
\[ (a+1)^2 = a^2 + 2a + 1, \quad (b+1)^2 = b^2 + 2b + 1, \quad (c+1)^2 = c^2 + 2c + 1 \]
- The third row becomes:
\[ (a-1)^2 = a^2 - 2a + 1, \quad (b-1)^2 = b^2 - 2b + 1, \quad (c-1)^2 = c^2 - 2c + 1 \]

Thus, the determinant can be rewritten as:

\[ D = \begin{vmatrix} a^2 & b^2 & c^2 \\ a^2 + 2a + 1 & b^2 + 2b + 1 & c^2 + 2c + 1 \\ a^2 - 2a + 1 & b^2 - 2b + 1 & c^2 - 2c + 1 \end{vmatrix} \]

### Step 2: Perform row operations

Next, we perform row operations to simplify the determinant. We can subtract the first row from the second and third rows:

- New second row:
\[ (a^2 + 2a + 1 - a^2, b^2 + 2b + 1 - b^2, c^2 + 2c + 1 - c^2) = (2a + 1, 2b + 1, 2c + 1) \]
- New third row:
\[ (a^2 - 2a + 1 - a^2, b^2 - 2b + 1 - b^2, c^2 - 2c + 1 - c^2) = (-2a + 1, -2b + 1, -2c + 1) \]

So, the determinant now looks like:

\[ D = \begin{vmatrix} a^2 & b^2 & c^2 \\ 2a + 1 & 2b + 1 & 2c + 1 \\ -2a + 1 & -2b + 1 & -2c + 1 \end{vmatrix} \]

### Step 3: Further simplify the determinant

Now we can further simplify by performing operations on the second and third rows. We can factor out constants from the second and third rows:

- Factor out 2 from the second row:
\[ D = 2 \begin{vmatrix} a^2 & b^2 & c^2 \\ a + \frac{1}{2} & b + \frac{1}{2} & c + \frac{1}{2} \\ -2a + 1 & -2b + 1 & -2c + 1 \end{vmatrix} \]

### Step 4: Evaluate the determinant

Now we can evaluate the determinant. If we set \( a = b = c \), we find that the determinant becomes 0. This means that the rows are linearly dependent, indicating that \( a, b, c \) must satisfy certain conditions.

### Conclusion

After performing the necessary operations and simplifications, we find that the determinant evaluates to zero, which implies that \( a, b, c \) must be equal. Therefore, \( a = b = c \) indicates that the triangle is equilateral.

Thus, the answer is that \( a, b, c \) are the sides of an equilateral triangle.