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Triangle-Closed Planar Sets

You are given a special triangle
T
(or a set of three non-collinear points) as a seed for a subset
S
of the plane.
For all pairs of points A, B in
S
you are asked to find the "closure point" C = C(
T
) such that the triangle ABC is similar to
T
.
Once C is found, we add it to the set
S
.
S
is called the
T
-closure or triangle-closure of
T
.
The cases where
S
is not a dense subset of the plane are the most interesting ones. Can you find a few?
Several examples of non-dense sets
S
are given (choose one from the "closed set" options).
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