There is a beautiful and interesting shape called the Arbelos - or cobble's knife. It consists of semicircle whose diameter is shared by the diameters of two other semicircles as shown.
Among the many interesting properties this beautiful shape possesses, one is a pair of circles called Archimedes's twins. They are two circles tucked tangentially between two semicircles and a common tangent to the smaller two as shown in \red.
Let O, C, D be the centers of the semicircles of radii respectively.
Let , r = center & radius of the Archimedes circle on the left. and be perpendicular to AB.
Then in triangle we can mark some distances as follows -
But we can also find MC as,
Equating (1) and (2) we get but Now we may derive the whole expression again for the Archimedes circle on the right or use symmetry of this expression to conclude it must hold for that circle too. Since the two circles have identical radii, they are called twins!