Can shapes determined by number of points?

From an amazing theorem in plane curves geometry we know that vertices of triangles similar to arbitrary triangle $T$ is **dense** on every closed jordan curve in a plane, so if $J$ be such curve and $A,B,C$ be 3 noncollinear points on plane then at least one curve similar to $J$ contains $A,B,C$. If $J$ be a circle then exactly one circle passes through $A,B,C$. From here we reach the problem in its simplest form:

**QUESTION:** is circle the only shape on a **Euclidean** plane (not only from closed curves mentioned above) which **just one similar to it** defined by each set of 3 non-collinear points we consider from $\mathbb{R}^2$ ? *(Means not two or more similar of the shape fits $A,B,C$, just one of it).*

**Note:** here the mentioned "shape" can be any subset of $\mathbb{R}^2$. The "trivial triangle" trivially could not be an answer because we can find many similar of it passing through the 3 points creates it as vertices. It seems for an specified set of 3 points with predefined angles of the triangle it creates, also we may find just circle which is unique.

What about generalization to $n$ points in **$\mathbb{R}^m$** which exactly define **$k$** similar shapes?