There are several statements known as "Miquel's Theorem", but the one I'm interested in here is Miquel's Six-Circle Theorem. It states that, if four points ABCD lie on a circle, and we draw four more circles through AB, BC, CD, and AD, the second intersection points of each of these four circles will also lie on a circle. Below I've drawn the first five circles in black, and the circle that must exist through the intersection points in red dashes.



One can find this in books and various web sources (e.g. here, here, or here). But what I haven't seen mentioned is that this can be interpreted in a very natural way as a statement about 3d geometry:

Theorem: Let a cuboid be placed in 3d in such a way that seven of its vertices belong to a sphere. Then the eighth vertex also belongs to the same sphere.

( Proof )

ETA, 1/23/08: Miquel's original paper can be found here, and already mentions the generalization to circles on a sphere, though not to vertices of cuboids. Thanks to Tim Robinson for digging all this up.