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 = b ne c `
C. ` a ne b = c`
D. ` a = b = 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-hand side
First, we will expand the right-hand side of the equation:
\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]

### Step 2: Substitute the expansion into the equation
Now, we substitute the expansion back into the equation:
\[ 3(a^2 + b^2 + c^2) = a^2 + b^2 + c^2 + 2(ab + ac + bc) \]

### Step 3: Rearrange the equation
Next, we will rearrange the equation to bring all terms to one side:
\[ 3(a^2 + b^2 + c^2) - (a^2 + b^2 + c^2) - 2(ab + ac + bc) = 0 \]
This simplifies to:
\[ 2(a^2 + b^2 + c^2) - 2(ab + ac + bc) = 0 \]

### Step 4: Factor out the common term
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 5: Rearranging gives us the final relation
Rearranging gives us:
\[ a^2 + b^2 + c^2 = ab + ac + bc \]

### Step 6: Recognize the relation
The equation \( a^2 + b^2 + c^2 = ab + ac + bc \) can be recognized as the condition for \( a, b, c \) to be the sides of an equilateral triangle. It also implies that:
\[ (a - b)^2 + (b - c)^2 + (c - a)^2 = 0 \]
This means \( a = b = c \).

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

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