Sets of circles

Question

I recently came up with a seemingly very easy problem I couldn't solve beyond the $ n =2 $ case.
Consider a set consisting of (the union of) $n+1$ circles (which might have different centers or radii). Show that if it is included (in terms of points) in a same kind of set with only $n$ circles, then at least two circles in the original set have same the center and radius. This seems really straightforward - I tried the pigeonhole principle to no avail.

Context: This "lemma" formalizes an idea I had to solve the following elementary problem (high-school level)

Show that the image of the set of nonzero complex numbers $z$ s.t. $\left|z+\frac{1}{z}\right|=2$ is the union of two circles, which shall be precised.

See Exercise 353 of https://www.louislegrand.fr/wp-content/uploads/2022/01/EXOS-TERMINALE3-3-AVECDESSIN.pdf (in French). As I couldn't find any solution for the initial problem on MSE (there's an unsatisfying/uneducational proof elsewhere where the author just drops the circles equations and checks the calculations), I might link to a typed version of mine when I have time.

Relevant MSE posts:

• $|z-1/z|=2$ where $z$ is a complex number. How to find the greatest value of $|z|$ geometrically?
• Find minimum $|z|$ satisfying $|z + 1/z |= 2$.

Answer 1

The intersection of any two different circles is a finite set of points, so a union of finitely many circles can only contain finitely many points of another circle that isn't one of the union members. Therefore, if a union of $n$ circles contains a whole circle $S$, then $S$ must be one of the $n$ original circles. The pigeonhole argument follows.