The three equal circles touch each other externally. If the centres of these circles be A, B, C then ABC is

Options

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

Answer

To solve the problem, we need to analyze the configuration of the three equal circles that touch each other externally. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have three equal circles, and we denote their centers as points A, B, and C.
- Since the circles touch each other externally, the distance between the centers of any two circles will be equal to the sum of their radii.

2. **Let the Radius be r**:
- Let the radius of each circle be \( r \).

3. **Calculating the Distances**:
- The distance between the centers A and B (denoted as AB) is equal to the sum of their radii:
\[ AB = r + r = 2r \]
- Similarly, the distance between centers B and C (denoted as BC) is:
\[ BC = r + r = 2r \]
- The distance between centers A and C (denoted as AC) is:
\[ AC = r + r = 2r \]

4. **Identifying the Triangle**:
- Now we have the lengths of all three sides of triangle ABC:
\[ AB = 2r, \quad BC = 2r, \quad AC = 2r \]
- Since all three sides are equal, triangle ABC is an equilateral triangle.

5. **Conclusion**:
- Therefore, the triangle ABC formed by the centers of the three equal circles is an equilateral triangle.

### Final Answer:
The triangle ABC is an equilateral triangle.