Pappus' Chain of Inscribed Circles
The chain of circles that fills an arbelos (shoemaker's knife) was first studied by the Greek mathematician, Pappus of Alexandria. It is not easy to construct the circles which must be tangent to adjacent circles and the circular arcs that make up the arbelos. A technique known a circle inversion is used to translate a chain of congruent circles (outside the arbelos) into the chain of circles inside the arbelos.
My inspiration to complete this task was a presentation by Hussein Tahir at the MAV (Maths Association of Victoria) annual conference in December 2012, the website from Florida Atlantic University (http://www.geogebra.org/en/examples/frisbee/index.html) and a paper by Ryan Young (http://www.morris.umn.edu/academic/math/Ma4901/Sp2012/Final/Ryan-Young-Final.pdf).