A problem with four circles and a square 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.

Hint
Let $A$ be the origin in the complex plane, and let $C$ be the point $c\in \mathbb{R}$
Let $M_0=re^{i \theta}$ so that $$M_1=\frac 12(c+M_0)$$
Then $$M_2=M_1+i(c-M_1)$$ and $$M_3=M_0+i(M_1-M_0)$$
Now you can obtain the parametric equations of the loci of $M_{1,2,3}$
