As is well-known, lines and circles are converted into lines and circles by circle inversion, or by any Möbius transformation for that matter. What bothers me is what happens in the Pappus's classical construction of a chain of circles inscribed into a region between two tangent circles (so-called Archimedes's arbelos).
One can apply a strategic circle inversion that converts this region into a strip between two parallel lines (see the first image on Mathhelp's Pappus Chain and below). Naturally, the inscribed circles are inverted into a vertical stack of circles inscribed into the strip, and their centers lie on its midline. The trouble is that the centers of the original Pappus chain circles lie on an ellipse (this is easy to show using its focal property, see e.g. Wikipedia's Pappus Chain). Since the inversion is involutive it would seem that it inverted a line into an ellipse?!
I am probably missing something very simple but I am not sure what. Is it that circle centers are not inverted into circle centers? Where does this ellipse go then?