Arbelos

The Arbelos is a plane figure bounded by three mutually tangent semicircles with collinear diameters. The outermost semicircle's diameter forms the base of the arbelos, and the two smaller semicircles are constructed internally on this base. The diameters of the two smaller semicircles sum to the diameter of the largest semicircle.

The word "arbelos" comes from the Greek word for "shoemaker's knife," due to its perceived resemblance to the blade used by shoemakers.

Notable properties of the arbelos include:

• Archimedes' Circles: Circles congruent to each other can be inscribed within the arbelos. Specifically, the twin circles of Archimedes are congruent circles tangent to each of the three semicircles that define the arbelos and to the line perpendicular to the base and passing through the point where the two inner semicircles meet. There are also an infinite number of other circles congruent to these Archimedes circles that can be inscribed within the arbelos, each tangent to the line separating the inner semicircles.

• Area: The area of the arbelos is equal to the area of a circle whose diameter is the line segment connecting the centers of the two inner semicircles.

• Pappus Chain: A chain of circles can be constructed within the arbelos, where each circle is tangent to both the base of the arbelos and to the two circles preceding it in the chain. These are known as the Pappus chain.

The arbelos has been studied extensively since antiquity, yielding many geometric relationships and theorems.