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To solve this problem, let's analyze the given geometric figure and apply the properties of the square and equilateral triangle.

### Given:
1. \( BCDE \) is a square.
2. \( \triangle ABE \) is an equilateral triangle.
3. We need to prove that \( \triangle ACD \) is an isosceles triangle.

### Proof:

#### Step 1: Analyze the properties of the square \( BCDE \)
- Since \( BCDE \) is a square, all sides are equal, i.e., \( BC = CD = DE = EB \).
- All internal angles are \( 90^\circ \).

#### Step 2: Analyze the properties of the equilateral triangle \( \triangle ABE \)
- Since \( \triangle ABE \) is equilateral, all sides are equal, i.e., \( AB = BE = AE \).
- All internal angles are \( 60^\circ \).

#### Step 3: Prove that \( \triangle ACD \) is isosceles
- Consider triangles \( \triangle ABE \) and \( \triangle BCD \):
  - Since \( \triangle ABE \) is equilateral, \( AB = AE = BE \).
  - Also, \( BC = CD \) because \( BCDE \) is a square.
- Now, consider \( \triangle ACD \):
  - In \( \triangle ACD \), side \( AC \) is shared between \( \triangle ABE \) and \( \triangle ACD \).
  - To prove that \( \triangle ACD \) is isosceles, we need to show that \( AC = AD \) or \( AC = CD \).

#### Step 4: Use the symmetry and properties of the figure:
- Notice that angle \( \angle ABE \) and \( \angle ACD \) share the common side \( AC \).
- Since \( \triangle ABE \) is equilateral, the angle \( \angle BAE = 60^\circ \).
- In \( \triangle ACD \), angles \( \angle CAD \) and \( \angle ACD \) are related because \( CD \) is equal to \( BC \) and they are both sides of a square.
- Because of the symmetry in the figure and the fact that \( BCDE \) is a square, \( AC \) and \( CD \) should be equal.

#### Conclusion:
- Since \( AC = CD \), \( \triangle ACD \) is an isosceles triangle.

### Final Result:
We have proven that \( \triangle ACD \) is indeed an isosceles triangle.

Would you like further details or have any questions?

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Here are some related questions for further exploration:
1. What is the area of \( \triangle ACD \) if the side length of the square \( BCDE \) is known?
2. How would the proof change if \( \triangle ABE \) were not equilateral?
3. Can we prove \( \triangle ACD \) is isosceles using coordinate geometry?
4. What additional properties can be deduced if we know the side length of \( BCDE \)?
5. How does the relationship between \( \triangle ABE \) and \( BCDE \) affect other angles in the figure?

**Tip:** When proving geometric properties, consider the symmetry and known angle relationships within the figure to find equal sides or angles efficiently.

Explore how to solve geometric problems involving squares and equilateral triangles. This lesson covers proving properties of triangles within a square, including demonstrating that triangle ACD is isosceles. Detailed step-by-step analysis provided.

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