Twin circles

Specifically, let A, B, and C be the three corners of the arbelos, with B between A and C. Let D be the point where the larger semicircle intercepts the line perpendicular to the AC through the point B. The segment BD divides the arbelos in two parts. The twin circles are the two circles inscribed in these parts, each tangent to one of the two smaller semicircles, to the segment BD, and to the largest semicircle. Each of the two circles is uniquely determined by its three tangencies. Constructing it is a special case of the Problem of Apollonius. Alternative approaches to constructing two circles congruent to the twin circles have also been found. These circles have also been called Archimedean circles. They include the Bankoff circle, Schoch circles, and Woo circles. ==Properties==

Let a and b be the diameters of two inner semicircles, so that the outer semicircle has diameter a + b. The diameter of each twin circle is then :d=\frac{ab}{a+b}. Alternatively, if the outer semicircle has unit diameter, and the inner circles have diameters s and 1-s, the diameter of each twin circle is :d=s(1-s). \, The smallest circle that encloses both twin circles has the same area as the arbelos. ==See also==