Ptolemy's Theorem
The latest Numberphile video is a beautiful proof of Ptolemy’s Theorem, presented by Zvezdelina Stankova.
Zvezda’s proof was followed by a second video, which utilises Ptolemy’s Theorem to show two remarkable things about equilateral triangles and pentagons - well worth a look and a much shorter video.
There is a third video, posted to the Numberphile2 extra channel, which shows a little bit more about the method of inversion.
PS: The first video’s length and “epic-ness” is perhaps matched only by our “Epic CIrcles” from 2014, which funnily enough also heavily featured the use of inversion. That video was presented by Simon Pampena.