Let $P$ be the plane with a point at infinity. By plane, I mean the Euclidian plane, and therefore it has circles. A line is also a circle, though its center is at infinity. If $A\subset P$ has cardinal $|A|=3$, there exists a unique circle (possibly a line) containing $A$; let me denote it $\Gamma_A$.

Let me say that a subset $X$ of $P$ is circularly stable if it satisfies the following property:

For every subset $A,B\subset X$ with $|A|=|B|=3$ and $\Gamma_A\ne \Gamma_B$, the intersection of $\Gamma_A$ and $\Gamma_B$ is included in $X$.

Every non-colinear $X$ with $|X|\le4$ is circularly stable. A line or a circle is circularly stable, and $P$ itself is so too. If $X$ has a non-void interior, then $X=P$.

Q: What can look like a circularly stable subset $X$ of the plane ? For instance, is it true that if $|X|\ge5$ and $X$ is circularly stable, then $X$ is dense in $P$, or $X$ is a line or a circle ?

Motivation: In classical geometry, one may wander what are the continuous maps $f:P\rightarrow P$ which transform circles or lines into circles or lines. If the guess above is true, then $f$ is completely determined by the images of $5$ non-colinear points.