# Archimedes' twin circles

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The Archimedes' circles (red) of an arbelos (grey)

In geometry, specifically in the study of the arbelos, the Archimedes' circles are two special circles associated with it.

Specifically, let $A$, $B$, and $C$ be the three corners of the arbelos, with $B$ between $A$ and $C$. Let $H$ be the point where the larger semicircle intercepts the line perpendicular to the $AC$ through the point $B$. The segment $BH$ divides the arbelos in two parts. The Archimedes' circles are the two circles inscribed in these parts, each tangent to one of the two smaller semicircles, to the segment $BH$, and to the largest semicircle.[1]

These circles are named after the Greek mathematician Archimedes, who defined them and showed that they are congruent, whatever the sizes of the semicircles $AB$ and $BC$. This is proposition 5 of his Book of Lemmas.[2] The circles are also known as the Archimedean circles, Archimedean twins, or other similar names.[3]

## Construction

Each of the two circles is uniquely determined by its three tangencies. Constructing it is a special case of the Problem of Apollonius.

## 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 Archimedean circle is then[1]

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

Alternatively, if the outer semicircle has unit diameter, and the inner circles have diameters $s$ and $1-s$, the diameter of each Archimedean circle is[1]

$d=s(1-s)$.

The smallest circle that encloses both Archimedean circles has the same area as the arbelos.[1]

## Alternative constructions

Since Archimedes, dozens of alternative ways of constructing two circles congruent to the Archimedean twins have been found,[4] and the list is still growing.[3][5]

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