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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

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 ?

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