This question is a follow up of my previous one (Planar sets closed under intersection of circles, Planar sets closed under intersection of circles) and is motivated by G. Zaimi's answer http://mathoverflow.net/questions/97010.

We are interested in subsets $X$ of the plane $P={\mathbb R}^2\cap{\infty}$, where ${\mathbb R}^2$ is Euclidian. We consider the property that for every $A,B\subset X$ of cardinal $3$, the circles passing through $A$ and $B$ either coincide or have their intersection contained in $X$. In this definition, a line is a circle passing through $\infty$. Note that if $X$ has this property, and if $M$ is a Moebius map in $P$, then $MX$ satisfies the same property.

It turns out that if $|X|\ge7$, then either $X$ is contained in a line or $X$ is dense in $P$. An interesting situation comes when $|X|=6$. Then, sending one point to $\infty$ by a Moebius map, the picture becomes a square together with its center and the point at infinity. The center can be sent to the origin by a translation, and then one vertex can be sent to $1$ by a similitude. Then $X$ is completely determined, the other points being $\pm i$ and $-1$.

My question is three-fold. First understand what subgroup of Moebius transforms leaves a given $X$ invariant; we may restrict to $X=(\infty,0,\pm1,\pm i)$. Second, every triplet $(a,b,c)$ belongs to finitely many such configurations $X$; but, how many ? Last, we may define a relation over the set $Q_3$ of triplets $(a,b,c)$ of pairwise distincts points in $P$, by $(a,b,c){\mathcal R}(d,e,f)$ if $X=(a,b,c,d,e,f)$ satisfies the property. Is this relation a classical object ?

This question is a follow up of my previous one (Planar sets closed under intersection of circles, Planar sets closed under intersection of circles) and is motivated by G. Zaimi's answer https://mathoverflow.net/questions/97010.

We are interested in subsets $X$ of the plane $P={\mathbb R}^2\cap{\infty}$, where ${\mathbb R}^2$ is Euclidian. We consider the property that for every $A,B\subset X$ of cardinal $3$, the circles passing through $A$ and $B$ either coincide or have their intersection contained in $X$. In this definition, a line is a circle passing through $\infty$. Note that if $X$ has this property, and if $M$ is a Moebius map in $P$, then $MX$ satisfies the same property.

It turns out that if $|X|\ge7$, then either $X$ is contained in a line or $X$ is dense in $P$. An interesting situation comes when $|X|=6$. Then, sending one point to $\infty$ by a Moebius map, the picture becomes a square together with its center and the point at infinity. The center can be sent to the origin by a translation, and then one vertex can be sent to $1$ by a similitude. Then $X$ is completely determined, the other points being $\pm i$ and $-1$.

My question is three-fold. First understand what subgroup of Moebius transforms leaves a given $X$ invariant; we may restrict to $X=(\infty,0,\pm1,\pm i)$. Second, every triplet $(a,b,c)$ belongs to finitely many such configurations $X$; but, how many ? Last, we may define a relation over the set $Q_3$ of triplets $(a,b,c)$ of pairwise distincts points in $P$, by $(a,b,c){\mathcal R}(d,e,f)$ if $X=(a,b,c,d,e,f)$ satisfies the property. Is this relation a classical object ?