Napoleon's Problem

I've been having quite a lot of fun playing with geometric constructions lately, and I have another one for you all today: a solution to Napoleon's Problem (that is, inscribing a square in a given circle using only a pair of compasses).

It's a famously tricky problem, and to be honest, I came up with this solution by accident while exploring something else. Nonetheless, I suspect this solution—involving six circles—is the simplest one possible, but I haven't managed to prove it yet.