It is asking me for the inversion of two concentric circles, but I just have no idea how to solve this, I've been stuck for a while
2026-03-29 08:44:37.1774773877
Proof that the composition of two inversion with the same center is a homothety
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Take a point $P$ on the plane, within distance $d$ from the common center. The first inversion takes $P$ to a point $Q$ within distance $d'=r_1^2/d$ from the center. The second inversion takes $Q$ to a new point $R$ a distance $d''=r_2^2/d'=r_2^2d/r_1^2$ from the center. By definition, the center, $P$, $Q$ and $R$ are aligned. The composition takes thus $P$ to a point on $OP$ within a distance $d''=kd$ from the center, where $k=r_2^2/r_1^2$. This is a homothety by definition.