NOTE: I have transcribed and edited this from various ancient translations of Euclid, augmented and tempered where necessary by at least the structure of more modern versions. I am pretty sure there are no errors in it. This is of course only one of many interesting Euclidean propositions involving hexagons, and for anyone reading this who does not in fact own a copy of Euclid I highly recommend purchasing one right now. Thank you.

A hexagon inscribed in a circle Let ABCDEF be the given circle. It is required to inscribe an equilateral and equiangular hexagon in the circle ABCDEF.