Construct an equilateral triangle with the given vertex so that the other vertices lie on the concentric circles respectively.
I constructed the triangle, but I don't know how it works. How does this construction work? Is there any proof?
Let the smaller circle be $a$, the larger circle $b$, and the point $c$.
Step 1: Construct a circle with radius of $b$ at the point $c$.
Step 2: The circle will intersect circle $a$ at $2$ points. Let the two points be $x$ and $y$. Construct a perpendicular bisector of the line connecting $x$ and the common centre of circle $a$ and $b$.
Step 3: The bisector intersect the circle $a$ at a point which is another vertex of the equilateral triangle.