For my college geometry class, I have to do a write up of a proof of the Butterfly Theorem. I'm doing a write up of the first proof given on that Page. It's pretty easy, but there's one thing that's bothering me. I know that inscribed angles subtended by the same chord are equal, but I can't find any theorem anywhere that says this. It isn't something that has been specifically stated in our class, and I'd rather say that this is true because of a theorem instead just stating that fact and not backing it up.

Does anyone here know what theorem this is, or any theorem that exists that has this fact as a corollary?

I found something that came from Euclid's Elements that says, "In a circle the angles in the same segment equal one another." Is that basically what I'm looking for, but with just different wording?

Also, I'm not quite clear on how they justify this step:

AX·XD/ CY·YB = PX·XQ/ PY·YQ

Thanks for all your help.