TITLE: Where do you see cyclic quadrilaterals in real life?
QUESTION [7 upvotes]: I've just been studying cyclic quads in geometry at school and I'm thinking see seems pretty interesting, but where would I actually find these in the real world? They seem pretty useless to me...

REPLY [7 votes]: Theorem 3 of the Bern-Eppstein paper cited below proves that any polygon of $n$ vertices
may be partitioned into $O(n)$ cyclic quadrilaterals.
A hint of how this might be achieved can be glimpsed in the figure below, where
all the white quadrilaterals are cyclic.

          


          

Fig.5 from cited paper.


Quadrilateral meshing is important in many applications. The cyclic
quads produced by their algorithm have desirable "quality" characteristics.

Bern, Marshall, and David Eppstein. "Quadrilateral meshing by circle packing." International Journal of Computational Geometry & Applications 10.04 (2000): 347-360. (Pre-journal arXiv abstract.)
  (Journal link.)