Triangle ABC is equilateral.
MA/ME is the golden ratio!
(George Hart showed me this result some time ago.)
Vynce Montgomery pointed out that therefore, this could be a way to construct a pentagram. To respond to this challenge, I hid everything but MA and ME, and proceeded from there:
PS: thinking about this a few days later, it occurs to me that if you don’t feel like doing the algebra, the second figure is pretty good confirmation of George’s claim.