TITLE: Proof of Miquel's six circle theorem
QUESTION [7 upvotes]: Theorem
Miquel's six circle theorem states that if in the following all cocircularities except the last one are satisfied, then the last one is implied.

In words: if $ABCD$ lie on a circle, and $ABYZ,BCXY,CDWX,DAZW$ likewise, then $XYZW$ lie on a circle (or a line) as well.
Motivation
The Wikipedia section on this is really short and does reference some books, but no online resources. So I think it would be nice to have various proofs for this theorem available here. I'll provide one (moved to an answer so it can be vote-sorted, too), but I want to know how other people with a different background would tackle this. So I make this an alternative-proof question.

REPLY [2 votes]: Since the motivation of the question was the lack of online resources, the theorem in question is Théorème I in  Miquel's paper Théorèmes sur  les  intersections  des  cercles  et  des  sphères, Journal de mathématiques pures et appliquées v3 (1838), 517-522.
It's proven via angle chasing.  As a reading note, the symbol $d$ refers to a right angle, so $2d$ is the angle $\pi$.  The figures are at "Planche III"
Theorem IV in the same paper states that stereographic projection takes circles on a sphere to circles on a plane.  The Ostermann-Wanner book claims this is a rediscovery of a theorem by Ptolmey.
The same volume also contains another paper by Miquel.