If `3 (a ^(2) + b ^(2) + c ^(2)) = (a + b + c) ^(2),` then the relation between a,b and c is.

Options

A. ` a ne b ne c`
B. `a ne b =c`
C. `a = b = c`
D. `a = b ne c`

Answer

To solve the equation \(3(a^2 + b^2 + c^2) = (a + b + c)^2\), we will follow these steps:

### Step 1: Expand the Right Side
First, we expand the right-hand side of the equation:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]
So, we rewrite the equation as:
\[ 3(a^2 + b^2 + c^2) = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]

### Step 2: Move All Terms to One Side
Next, we will move all terms to one side of the equation:
\[ 3(a^2 + b^2 + c^2) - (a^2 + b^2 + c^2) - 2(ab + ac + bc) = 0 \]
This simplifies to:
\[ 2(a^2 + b^2 + c^2) - 2(ab + ac + bc) = 0 \]

### Step 3: Factor Out Common Terms
Now, we can factor out the common term of 2:
\[ 2(a^2 + b^2 + c^2 - ab - ac - bc) = 0 \]
Since 2 is not zero, we can simplify this to:
\[ a^2 + b^2 + c^2 - ab - ac - bc = 0 \]

### Step 4: Rearranging the Equation
Rearranging gives us:
\[ a^2 + b^2 + c^2 = ab + ac + bc \]

### Step 5: Recognizing the Condition
The equation \(a^2 + b^2 + c^2 = ab + ac + bc\) holds true if and only if \(a = b = c\).

### Conclusion
Thus, the relation between \(a\), \(b\), and \(c\) is:
\[ a = b = c \]