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.