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 :
If `angleA=60^(@)`, then `Delta ABC` is

Options

A. isoceles
B. right angled
C. right angled isosceles
D. equilateral

Explanation

To solve the problem, we will analyze the given conditions and apply the relevant properties of triangles.

### Step-by-Step Solution:

1. **Given Information**: We have triangle ABC where the angle A = 60°. The line joining the circumcenter (O) and incenter (I) is parallel to the base BC.

2. **Understanding the Relationship**: The property we will use is that the ratio of the inradius (r) to the circumradius (R) is given by:
\[ \frac{r}{R} = \cos A \]
Here, since angle A = 60°, we have:
\[ \cos 60° = \frac{1}{2} \]
Therefore, we can write:
\[ \frac{r}{R} = \frac{1}{2} \]

3. **Analyzing the Ratio**: The ratio \(\frac{r}{R} = \frac{1}{2}\) indicates that this is the maximum value for the ratio of the inradius to the circumradius. In general, the ratio of the inradius to the circumradius for any triangle is always less than or equal to \(\frac{1}{2}\).

4. **Conclusion about the Triangle**: The condition \(\frac{r}{R} = \frac{1}{2}\) occurs specifically when the triangle is equilateral. Since we have established that this ratio is at its maximum, we conclude that triangle ABC must be equilateral.

5. **Final Answer**: Therefore, triangle ABC is an equilateral triangle.