Three circles with equal radii touch each other externally. The figure formed by joining the centres of these circles is

Options

A. an isosceles triangle
B. an equilateral triangle
C. a scalene triangle
D. a right angled triangle

Answer

To solve the given problem, we need to analyze the situation of three circles with equal radii that touch each other externally. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Circles**: We have three circles, all with the same radius \( r \). They are positioned such that each circle touches the other two externally.

2. **Positioning the Circles**:
- Place the first circle at a point, say \( A \).
- The second circle touches the first circle at point \( B \).
- The third circle touches both the first and the second circles at points \( C \) and \( D \) respectively.

3. **Connecting the Centers**:
- Let the centers of the three circles be \( O_1 \), \( O_2 \), and \( O_3 \).
- The distance between any two centers (for example, \( O_1 \) and \( O_2 \)) is equal to the sum of their radii since they touch each other externally. Therefore, the distance \( O_1O_2 = r + r = 2r \).

4. **Calculating Distances**:
- Similarly, the distance \( O_2O_3 = 2r \) and \( O_3O_1 = 2r \).
- Thus, all sides of the triangle formed by connecting the centers \( O_1 \), \( O_2 \), and \( O_3 \) are equal to \( 2r \).

5. **Identifying the Figure**:
- Since all three sides of the triangle formed by the centers are equal, the figure is an equilateral triangle.

### Conclusion:
The figure formed by joining the centers of the three circles is an **equilateral triangle**.