Let ABC be a triangle in which the line joining the circumecentre and incentre is parallel to base BC of the triangle. Then answer the following questions :
Then range of `angle A` is

Options

A. `[(pi)/(6),(pi)/(3)]`
B. `[(pi)/(3),(pi)/(2))`
C. `[(pi)/(3),(2pi)/(3)]={(pi)/(3)}`
D. `[0,(pi)/(2)]`

Explanation

To solve the problem, we need to find the range of angle A in triangle ABC, given that the line segment joining the circumcenter (O) and the incenter (I) is parallel to the base BC.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have triangle ABC with circumcenter O and incenter I.
- The line segment OI is parallel to the base BC.

2. **Properties of Parallel Lines**:
- Since OI is parallel to BC, we can conclude that the distances from O to line BC and from I to line BC are equal. This implies that the quadrilateral OIDE (where D is the foot of the perpendicular from O to BC, and E is the foot of the perpendicular from I to BC) is a rectangle.

3. **Using the Properties of the Triangle**:
- In this rectangle, we have:
- OD = IE
- Here, OD can be expressed in terms of angle A as:
- \( OD = r \cos A \) (where r is the circumradius).
- The length IE is simply the inradius \( r \).

4. **Setting Up the Equation**:
- Since OD = IE, we have:
\[ r \cos A = r \]
- Dividing both sides by r (assuming r > 0), we get:
\[ \cos A = 1 \]

5. **Understanding the Range of Cosine**:
- The cosine function varies between -1 and 1. However, for angles in a triangle, we restrict our attention to the range where \( \cos A \) is non-negative.
- Therefore, we have:
\[ 0 < \cos A \leq \frac{1}{2} \]

6. **Finding the Corresponding Angles**:
- The angle A corresponds to:
- \( \cos A = 0 \) at \( A = 90^\circ \) (or \( \frac{\pi}{2} \) radians).
- \( \cos A = \frac{1}{2} \) at \( A = 60^\circ \) (or \( \frac{\pi}{3} \) radians).

7. **Conclusion**:
- Therefore, the range of angle A is:
\[ \frac{\pi}{3} < A < \frac{\pi}{2} \]

### Final Answer:
The range of angle A is \( \left( \frac{\pi}{3}, \frac{\pi}{2} \right) \).