If `2s = a + b + c`, then the value of
`(s - a)^2 + (s - b)^2 + (s - c)^2 + s^2 - a^2 - b^2 - c^2` will be :

Options

A. `-1`
B. `1`
C. `2`
D. `0`

Answer

To solve the expression \( (s - a)^2 + (s - b)^2 + (s - c)^2 + s^2 - a^2 - b^2 - c^2 \) given that \( 2s = a + b + c \), we can follow these steps:

### Step 1: Expand the squares
We start by expanding the squares in the expression:

\[ (s - a)^2 = s^2 - 2sa + a^2 \]
\[ (s - b)^2 = s^2 - 2sb + b^2 \]
\[ (s - c)^2 = s^2 - 2sc + c^2 \]

Now, we can substitute these expansions into our original expression:

\[ (s - a)^2 + (s - b)^2 + (s - c)^2 = (s^2 - 2sa + a^2) + (s^2 - 2sb + b^2) + (s^2 - 2sc + c^2) \]

### Step 2: Combine like terms
Combining all the terms, we have:

\[ = 3s^2 - 2s(a + b + c) + (a^2 + b^2 + c^2) \]

### Step 3: Substitute \( a + b + c \)
From the problem, we know that \( 2s = a + b + c \). Therefore, we can substitute \( a + b + c \) with \( 2s \):

\[ = 3s^2 - 2s(2s) + (a^2 + b^2 + c^2) \]
\[ = 3s^2 - 4s^2 + (a^2 + b^2 + c^2) \]
\[ = -s^2 + (a^2 + b^2 + c^2) \]

### Step 4: Add the remaining terms
Now we add the remaining part of the expression, which is \( s^2 - a^2 - b^2 - c^2 \):

\[ -s^2 + (a^2 + b^2 + c^2) + s^2 - a^2 - b^2 - c^2 \]

### Step 5: Simplify the expression
Combining these terms:

\[ = (-s^2 + s^2) + (a^2 - a^2) + (b^2 - b^2) + (c^2 - c^2) \]
\[ = 0 \]

Thus, the value of the expression is \( 0 \).

### Final Answer
The value of the expression is \( 0 \).

---