I was doing random things when I noticed something which seemed strange to me.
What I did was the following.
Take two points $A$ and $C$ of the plane. We denote $\ell$ the distance $AC$.
Draw a circle $\mathscr C_0$ of center $A$ and radius $r_0$.
Then define $M_0$ to be a point of $\mathscr C_0$, and $M_1$ the middle of $[M_0C]$.
Finally, define $M_2$ and $M_3$ such that $M_0M_1M_2M_3$ is square.
It should look like something like this:
Then we are interested in the trajectory of $M_1$, $M_2$ and $M_3$ when $M_0$ move along the circle $\mathscr C_0$.
We notice that every $M_i$ seems to move on a circle $\mathscr C_i$ of a unique radius $r_i$.
But I don't get why this would be true, which leads us to the first questions:
Question 1. Are all trajectories $\mathscr C_i$ circles?
Question 2. What are the radius $r_i$ in terms of $\ell$ and $r_0$?
Question 3. Where are located the centers of those circles?
What I noticed is that if the radius $r_0$ changes, we still get three other circles, and they are all concentric:
And this is the case even when $r_0\geqslant \ell$:
It looks like this if $r_0$ varies continuously:
What I did to try to find the centers (since all three circles seems to have the same three centers for different radius $r_0$) is to see what it would look like with $r_0$ really small:
So the centers seems to be:
the middle point $A_1$ of $[AC]$,
the two points $A_2$, $A_3$ such that $AA_1A_2A_3$ is a square.