In a triangle ABC if the angles A, B, C are in AP, then which one of the following is correct ?

Options

A. c = a + b
B. `c^(2) = a^(2) + b^(2) - ab`
C. `a^(2) = b^(2) + c^(2) - bc`
D. `b^(2) = a^(2) + c^(2) - ac`

Explanation

To solve the problem, let's follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem:**
We have a triangle ABC where the angles A, B, and C are in Arithmetic Progression (AP). We need to find a relationship involving the sides of the triangle based on this condition.

2. **Using the Property of Angles in AP:**
If angles A, B, and C are in AP, we can express this as:
\[ 2B = A + C \]
This means that angle B is the average of angles A and C.

3. **Using the Sum of Angles in a Triangle:**
The sum of the angles in a triangle is always 180 degrees:
\[ A + B + C = 180^\circ \]

4. **Substituting for A + C:**
From the first equation \(2B = A + C\), we can substitute \(A + C\) in the sum of angles equation:
\[ A + C = 2B \implies 2B + B = 180^\circ \]
This simplifies to:
\[ 3B = 180^\circ \]

5. **Solving for B:**
Dividing both sides by 3 gives:
\[ B = 60^\circ \]

6. **Finding Angles A and C:**
Now, we can find angles A and C using the relationship \(A + C = 2B\):
\[ A + C = 2 \times 60^\circ = 120^\circ \]
Since \(A + B + C = 180^\circ\), we can also express:
\[ A + C = 180^\circ - B = 180^\circ - 60^\circ = 120^\circ \]
This confirms our earlier calculation.

7. **Using the Law of Cosines:**
We can relate the sides of the triangle to the angles using the Law of Cosines:
\[ \cos B = \frac{A^2 + C^2 - B^2}{2AC} \]
Substituting \(B = 60^\circ\):
\[ \cos 60^\circ = \frac{1}{2} \]
Therefore, we have:
\[ \frac{A^2 + C^2 - B^2}{2AC} = \frac{1}{2} \]

8. **Cross-Multiplying:**
Cross-multiplying gives:
\[ A^2 + C^2 - B^2 = AC \]

9. **Rearranging the Equation:**
Rearranging this equation leads us to:
\[ B^2 = A^2 + C^2 - AC \]

10. **Conclusion:**
Thus, the correct relationship involving the sides of the triangle is:
\[ B^2 = A^2 + C^2 - AC \]

### Final Answer:
The correct option is \(B^2 = A^2 + C^2 - AC\).