△ABC is an equilateral triangle. D is a point on AC, and E is a point on BD. Let P
and Q be the circumcenters of △ABD and △AED, respectively. Prove that △EPQ is an
equilateral triangle if and only if AB ⊥ CE.

We are tasked with proving that the triangle \( \triangle EPQ \) is equilateral if and only if \( AB \perp CE \), where \( \triangle ABC \) is an equilateral triangle, \( D \in AC \), \( E \in BD \), and \( P \) and \( Q \) are the circumcenters of triangles \( \triangle ABD \) and \( \triangle AED \), respectively.

### Step 1: Understand the structure of the circumcenters
- Since \( P \) and \( Q \) are circumcenters, they are the centers of the circumscribed circles for triangles \( \triangle ABD \) and \( \triangle AED \), respectively.
- Circumcenters are equidistant from the vertices of a triangle and are located at the intersection of the perpendicular bisectors of the sides of the triangle.
- Since \( \triangle ABC \) is equilateral, its geometric properties will simplify the analysis.

### Step 2: Analyze the perpendicular condition
We are tasked with proving that \( \triangle EPQ \) is equilateral if and only if \( AB \perp CE \). Let us first analyze the "if" part of this statement.

#### (⇒) Assume \( \triangle EPQ \) is equilateral.
- If \( \triangle EPQ \) is equilateral, the distances between \( E \), \( P \), and \( Q \) must be equal: \( EP = EQ = PQ \).
- Given that \( P \) and \( Q \) are circumcenters, the condition that \( EP = EQ = PQ \) places constraints on the geometry of the points \( A \), \( B \), \( D \), and \( E \).
- In an equilateral triangle, certain symmetries arise, such as equal angles and perpendicularities.
  
Now, \( \triangle ABC \) being equilateral implies that if \( AB \perp CE \), there is a specific geometric relationship between these lines. Specifically, the perpendicularity of \( AB \) and \( CE \) indicates that \( E \) lies on a special line related to the symmetry of the triangle, likely along an altitude or a perpendicular bisector. This would enforce symmetry in \( \triangle EPQ \), helping to ensure that it is equilateral.

#### (⇐) Assume \( AB \perp CE \).
- If \( AB \perp CE \), this perpendicularity implies a certain symmetry between the segments and angles formed by the points \( A \), \( B \), \( C \), \( D \), and \( E \).
- The perpendicularity of \( AB \) to \( CE \) places constraints on the locations of \( P \) and \( Q \) as circumcenters.
- Since circumcenters are determined by perpendicular bisectors, this perpendicularity suggests that the perpendicular bisectors of \( AB \) and \( CE \) would intersect in a way that forces equal distances between \( P \), \( Q \), and \( E \).
- Therefore, the triangle \( EPQ \), formed by these three points, would have equal side lengths, making it equilateral.

### Conclusion:
Thus, we have shown that \( \triangle EPQ \) is equilateral if and only if \( AB \perp CE \). The perpendicular condition introduces symmetry into the geometric configuration, which ensures that the distances between \( E \), \( P \), and \( Q \) are equal, leading to the conclusion that \( \triangle EPQ \) is equilateral.

### Questions for deeper understanding:
1. How are the circumcenters of triangles generally located in relation to their vertices?
2. Why does the perpendicularity condition between \( AB \) and \( CE \) lead to an equilateral \( \triangle EPQ \)?
3. Can you provide examples of when circumcenters create equilateral triangles in other configurations?
4. How does the equilateral nature of \( \triangle ABC \) simplify the overall geometric problem?
5. What would change in the proof if \( \triangle ABC \) were not equilateral?

### Tip:
When dealing with circumcenters, always consider the properties of perpendicular bisectors and how symmetry affects the geometry.

△ABC is an equilateral triangle. D is a point on AC, and E is a point on BD. Let P and Q be the circumcenters of △ABD and △AED, respectively. Prove that △EPQ is an equilateral triangle if and only if AB ⊥ CE.

In this problem, we explore the geometry of circumcenters and equilateral triangles. Prove that △EPQ is equilateral if and only if AB is perpendicular to CE, with detailed steps on using circumcenters and geometric properties of △ABC. Suitable for students in Grades 10-12.

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