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Determine all bounded closed subsets

F

of the plane with the following property:

F

consists of at least two points and always contains two points

A

and

B

as well as at least one of the two semicircular arcs over the segment

AB

.
Definitions: A subset of the

F

of the plane is said to be closed if: For every point

P

of the plane that is not an element of

F

, there is a (non-degenerate) disc with center

P

that has no elements of

F

Problem Statement