# arbelos

An arbelos is the figure bounded by three semicircles *AB*, *BC*,
and *AC*, where *ABC* is a straight line. Archimedes (about 250 BC) called it an "arbelos" – the Greek word for a knife
of the same shape used by shoemakers to cut and trim leather – and
wrote about it in his *Liber assumptorum* (Book of Lemmas).

Among the properties of the arbelos are that the sum of the two smaller
arc lengths is equal to the larger; the area of the arbelos is π/4 times
the product of the two smaller diameters (*AB* and *BC*); and
the area of the arbelos is equal to the area of a circle whose diameter is the length of a perpendicular segment drawn from the tangent
point *B* of the two smaller semicircles to the point *D* where
it meets the larger semicircle.

The circles inscribed on each half of *BD* of the arbelos (called **Archimedes's
circles**) each have a diameter of (*AB*)(*BC*)/(*AC*).
Furthermore, the smallest circumcircle of these two circles has an area
equal to that of the arbelos.

Pappus of Alexandria wrote on the relations
of the chain of circles, *C*1, *C*2, *C*3, ..., (called a **Pappus chain** or an **arbelos train**) that
are mutually tangent to the two large semicircles and to each other. The
centers of these circles lie on an ellipse and the diameter of the *n*th
circle is (1/*n*) times the base of the perpendicular distance to the
base of the semicircle.