D, E and F are the mid-points of the sides AB, BC and CA of an isosceles triangle ABC in which AB = BC. Prove that `DeltaDEF` is also isosceles.

Answer

To prove that triangle DEF is isosceles, we will use the properties of midpoints and the Midpoint Theorem. Let's go through the steps systematically.

### Step-by-Step Solution:

1. **Identify the Triangle and Midpoints**:
- Let triangle ABC be an isosceles triangle where AB = BC.
- Let D, E, and F be the midpoints of sides AB, BC, and CA respectively.

2. **Apply the Midpoint Theorem**:
- According to the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
- Therefore, since D and E are midpoints of AB and BC:
\[ DE = \frac{1}{2} AC \]

3. **Find Lengths of Other Sides**:
- Similarly, since E and F are midpoints of BC and CA:
\[ EF = \frac{1}{2} AB \]
- And since F and D are midpoints of CA and AB:
\[ FD = \frac{1}{2} BC \]

4. **Use the Given Information**:
- Since AB = BC (as given in the problem), we can substitute:
\[ EF = \frac{1}{2} AB = \frac{1}{2} BC \]

5. **Establish Equality of Two Sides**:
- Now we have:
\[ DE = \frac{1}{2} AC \quad \text{and} \quad EF = \frac{1}{2} AB \]
- Since AB = BC, we also have:
\[ EF = \frac{1}{2} BC \]
- Thus, we can conclude:
\[ DE = EF \]

6. **Conclusion**:
- Since DE = EF, triangle DEF is isosceles with DE = EF.

### Final Statement:
Therefore, we have proved that triangle DEF is isosceles.

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