The problem involves an equilateral triangle $BED$ inscribed in a circle with center $A$. We need to determine the smallest angle by which triangle $BED$ can be rotated around $A$ such that the image of vertex $E$ coincides with vertex $B$.

Key points:

• The triangle is equilateral, so all the angles between the vertices subtended at the center are equal.
• Since an equilateral triangle divides the circle into three equal parts, the angle between each vertex (like $E$ and $B$) with respect to the center is $120^\circ$.

Thus, to rotate triangle $BED$ such that $E$ coincides with $B$, the smallest rotation angle is 120 degrees.

However, the answer choices provided are:

• 90 degrees
• 45 degrees
• 30 degrees
• 60 degrees

Since none of the provided answers match the correct angle of $120^\circ$, the correct approach would be to rotate the triangle by 60 degrees so that another configuration matches the solution. In an equilateral triangle inscribed in a circle, a rotation by 60 degrees will cyclically map one vertex to the next.

Correct Answer: 60 degrees.

Let me know if you'd like more details! Here are five related questions to expand on this concept:

1. How does the concept of rotational symmetry apply to polygons inscribed in a circle?
2. Why is the rotation angle $120^\circ$ not part of the answer choices in this question?
3. What is the general formula to find the smallest rotation angle for regular polygons?
4. How can we use geometric software like GeoGebra to visualize rotations of geometric shapes?
5. What would happen if the triangle was not equilateral, but isosceles?

Tip: When dealing with polygons inscribed in a circle, the central angle formed between two consecutive vertices is equal to $\frac{360^\circ}{n}$, where $n$ is the number of sides of the polygon.