Jiřka constructed two congruent equilateral triangles as shown in the figure. She further wants to construct all circles that will have their center at one of the vertices and will pass through another vertex of one of the triangles.

Construct and calculate all circles that meet Jiřka's requirements.

(K. Pazourek)

![](https://cdn.mathpix.com/cropped/2024_04_17_a44687afde4ea037c0fdg-4.jpg?height=420&width=257&top_left_y=1709&top_left_x=905)

Hint. Start with the vertices that are common to both triangles.
Let's name the vertices as shown in the following image and note that points $A$ and $C$ are always equidistant from the remaining three points (the corresponding segments form the sides of equilateral triangles). Therefore, the circle centered at point $A$ passing through point $B$ also passes through points $C$ and $D$. The circle centered at point $A$ that meets Jiřčin's requirements is thus unique. Similarly, there is a unique circle centered at point $C$ that meets the requirements.

The circle centered at point $B$ passing through point $A$ also passes through point $C$, and another circle passes through point $D$. Therefore, there are two circles centered at point $B$ that meet Jiřčin's requirements. Similarly, there are two circles centered at point $D$ that meet the requirements.

In total, there are six circles that meet the requirements:

![](https://cdn.mathpix.com/cropped/2024_04_17_a44687afde4ea037c0fdg-5.jpg?height=874&width=596&top_left_y=445&top_left_x=730)