Anyone know why this works?

https://www.youtube.com/shorts/M4jXRKgAutM

(Boosts gratefully received 🙏)

Circle Inscribing an Equilateral Triangle | Construction

YouTube

There may be an easier proof, but this was the best I could do. Would love to hear other ideas, or if there's anything I've missed 🙏

cc: @skua @ob1quixote @F100