Arbelos HINT: The height of the circle standing at the tangent point of circles C1 and C2 is the geometric mean of the diameters of the two circles: height = \(\sqrt(2\cdot r\cdot (1-2\cdot r))\).
Pappus Chain HINT: The height of the centre of the circle P1 lies \(2 \cdot radius\) of circle P1 above the x-axis.
The height of the centre of the next circle (P2) lies \(4 \cdot radius\) of that circle above the x-axis.
Pappus Ellipse HINT: The two focal points of the Pappus ellipse lie at the centres of the two circles which define the chain: the circle with centre 0 and circle C1. The centre of the ellipse lies halfway between these points.
\[(\frac{x-\frac{1}{4}(1+a)}{\frac{1}{4}(1+a)})^2+(\frac{y}{\frac{1}{2}\sqrt(a)})^2=1\]
To learn more, research the Pappus Chain further.