SIMO Retiree Blog

(This is David.) I'm back with a short post about a beautiful proof for a beautiful problem I saw recently. Three dimensions? Let me explain the title. I think it was during a decent IMO where Grant Sanderson (of 3blue1brown fame) gave a talk about problems that are super easy once we move to a higher dimension. If you weren't there at the talk, he also made it into a youtube video - I highly recommend watching it if you haven't already! Here at the SIMO X-Men blog, we aren't unfamilliar with this idea - one of the most popular blogposts to date is Glen's Spacetime, Special Relativity, and a Lot of Circles where we saw that interpreting circles as points in 3-dimensional space was a really powerful tool for lots of geometry problems involving tangent circles. And the nice thing is, this trick doesn't stop at puzzles and Olympiad problems - it also shows up in real research. Arguably, the recent breakthrough for the sofa problem used this idea, and I've...

 (This is Glen.) At some point, Ker Yang wrote a post on Reim's theorem, which he used to solve two past year IMO problems. I remember commenting to him that I had solved neither of them with Reim (and no, I did not bash them). Later on, I tried reconstructing my solution to IMO 2017/4, and I noticed something interesting that made me find another (slightly weird) solution that (I think) isn't on AoPS. So that's what I'll be writing about today.

(It's Glen again!) This post is going to be long, and as the title suggests, contain very many circles. Relatedly, most of the diagrams in here are hand-drawn and scanned because no way am I going to figure out how draw all of these diagrams on computer. You have been warned.