Twin circles

Specifically, let ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ be the three corners of the arbelos, with ${\displaystyle B}$ between ${\displaystyle A}$ and ${\displaystyle C}$. Let ${\displaystyle H}$ be the point where the larger semicircle intercepts the line perpendicular to the ${\displaystyle AC}$ through the point ${\displaystyle B}$. The segment ${\displaystyle BH}$ 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 ${\displaystyle BH}$, and to the largest semicircle.[3]

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.[4][5] These circles have also been called Archimedean circles. They include the Bankoff circle, Schoch circles, and Woo circles.

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[3]

${\displaystyle d={\frac {ab}{a+b}}.}$

Alternatively, if the outer semicircle has unit diameter, and the inner circles have diameters ${\displaystyle s}$ and ${\displaystyle 1-s}$, the diameter of each twin circle is[3]

${\displaystyle d=s(1-s).\,}$

The smallest circle that encloses both twin circles has the same area as the arbelos.[3]

• Schoch line