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).