ABC is an equilateral triangle. Points D, E, F are taken in sides AB, BC, CA respectively, so that AD = BE = CF. then AE, BF, CD enclosed a triangle which is:

Options

A. equilateral triangle
B. isosceles triangle
C. right angle triangle
D. none of these

Answer

To solve the problem, we need to analyze the given equilateral triangle ABC and the points D, E, and F on its sides. The goal is to determine the nature of the triangle formed by the segments AE, BF, and CD.

### Step-by-Step Solution:

1. **Understanding the Equilateral Triangle**:
- Let ABC be an equilateral triangle, meaning all sides are equal, and all angles are 60 degrees.

2. **Identifying Points D, E, and F**:
- Points D, E, and F are located on sides AB, BC, and CA respectively such that:
- AD = BE = CF = x (some equal length).

3. **Finding the Lengths of the Segments**:
- Since AD = x, then:
- BD = AB - AD = AB - x
- Similarly, for points E and F:
- BE = x, thus EC = BC - BE = BC - x
- CF = x, thus AF = CA - CF = CA - x

4. **Using the Properties of Equilateral Triangles**:
- In an equilateral triangle, since all sides are equal, we have:
- AB = BC = CA
- Therefore, we can denote the length of each side as 's'.
- Thus:
- BD = s - x
- EC = s - x
- AF = s - x

5. **Analyzing the Triangle Formed by AE, BF, and CD**:
- Now we calculate the lengths of AE, BF, and CD:
- AE = AB - AD = s - x
- BF = BC - BE = s - x
- CD = CA - CF = s - x
- Therefore, we have:
- AE = BF = CD = s - x

6. **Conclusion**:
- Since AE, BF, and CD are all equal, the triangle formed by these segments (triangle DEF) is also equilateral.

### Final Answer:
The triangle formed by segments AE, BF, and CD is an **equilateral triangle**.

---