△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 △EP Q is an
equilateral triangle if and only if AB ⊥ CE.

To prove that \(\triangle EPQ\) is an equilateral triangle if and only if \(AB \perp CE\), let's break down the problem into key steps:

### Step 1: Analyze Given Conditions and Construct the Problem
1. Let \(\triangle ABC\) be an equilateral triangle, so \(AB = BC = CA\) and all angles are \(60^\circ\).
2. \(D\) is a point on segment \(AC\) and \(E\) is a point on segment \(BD\).
3. Points \(P\) and \(Q\) are the circumcenters of \(\triangle ABD\) and \(\triangle AED\), respectively.
4. We want to show that \(\triangle EPQ\) is an equilateral triangle if and only if \(AB \perp CE\).

### Step 2: Understanding the Circumcenters and Their Properties
1. Since \(\triangle ABC\) is equilateral, any triangle formed within it, such as \(\triangle ABD\) and \(\triangle AED\), will have certain symmetry properties, especially when considering their circumcenters.
2. The circumcenter \(P\) of \(\triangle ABD\) is equidistant from \(A\), \(B\), and \(D\).
3. Similarly, the circumcenter \(Q\) of \(\triangle AED\) is equidistant from \(A\), \(E\), and \(D\).

### Step 3: Explore Conditions for \(\triangle EPQ\) to be Equilateral
To show that \(\triangle EPQ\) is equilateral, we need:
1. The lengths \(EP = PQ = QE\) to be equal.
2. Each angle of \(\triangle EPQ\) to measure \(60^\circ\).

For this configuration to hold, it is critical that:
- The symmetry of \(\triangle EPQ\) depends on the positioning of \(D\) and \(E\) in relation to \(AB\) and \(CE\).
- Specifically, if \(AB \perp CE\), then the circumcenters \(P\) and \(Q\) will align in such a way that \(\triangle EPQ\) forms an equilateral triangle.

### Step 4: Prove the "If and Only If" Condition
1. **If** \(AB \perp CE\): 
   - When \(AB \perp CE\), the perpendicularity imposes a rotational symmetry in the configuration, making the distances \(EP\), \(PQ\), and \(QE\) equal due to the equilateral nature of \(\triangle ABC\).
   - This perpendicular relationship ensures that \(\angle EPQ = 60^\circ\), forming an equilateral triangle \(\triangle EPQ\).

2. **Only If** \(\triangle EPQ\) is equilateral:
   - If \(\triangle EPQ\) is equilateral, then by symmetry, the perpendicularity \(AB \perp CE\) must hold. This is because any deviation from perpendicularity would disrupt the symmetry needed for all sides and angles of \(\triangle EPQ\) to be equal.

### Conclusion
Therefore, we have shown that \(\triangle EPQ\) is an equilateral triangle if and only if \(AB \perp CE\).

Would you like more details on any specific step or further explanation on circumcenters in this context?

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Here are some related questions to explore further:
1. What is the significance of circumcenters in equilateral triangles?
2. How does perpendicularity influence the properties of triangles within equilateral triangles?
3. Can we generalize this property to non-equilateral triangles?
4. What role does symmetry play in proving geometric theorems?
5. How can we use coordinate geometry to verify this proof?

**Tip:** When proving "if and only if" statements in geometry, try breaking it down into two separate proofs, one for each direction, to ensure a thorough understanding.

△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.

Explore a proof showing △EPQ is an equilateral triangle in △ABC if and only if AB ⊥ CE. Learn how circumcenters and perpendicularity impact this unique geometric configuration. This problem is ideal for high school geometry students covering equilateral triangles and circumcenters.

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