How can I use inversion in a circle to simplify a problem?

[esisk]

Can somebody give me an example whereby I use the inversion with respect to a circle (unit circle or otherwise) and the problem becomes easier. I guess I am asking: how do I make use of this notion. Or a problem that involves inversion, period.
Thank you

 

[HallsofIvy]

The only time I have used inversion in a circle was in Poincare's disk model for hyperbolic geometry. There "congruence" is defined in terms of reflections in a "line", "lines" are the portions of circles orthogonal to the disk inside the disk, and "reflection" in such a line is inversion in the circle.

In this article, http://en.wikipedia.org/wiki/Inversive_geometry, Wikipedia refers to using inversion in a circle to construct a "Peaucellier linkage", apparently important in "converting between linear and circular motion". I have heard that one can use inversion in a circle to model Wankel Rotary Engine but have no certain information on that.