The term "arbelos" means shoemaker's knife in Greek, and this term is applied to the shaded area in the above figure which resembles the blade of a knife used by ancient cobblers (Gardner 1979). Archimedes himself is believed to have been the first mathematician to study the mathematical properties of this figure. The position of the central notch is arbitrary and can be located anywhere along the diameter.
The arbelos satisfies a number of unexpected identities (Gardner 1979, Schoch).
1. Call the diameters of the left and right semicircles 
and 
, respectively, so the diameter of the enclosing semicircle
is 1. Then the arc length along the bottom of the arbelos is
![L=1/2[pir+pi(1-r)]=1/2pi,](/images/equations/Arbelos/NumberedEquation1.svg) |
(1)
|
so the arc length along the enclosing semicircle is the same as the arc length along the two smaller semicircles.
2. Draw the perpendicular 
from the tangent of the two semicircles
to the edge of the large circle. Then the area
of the arbelos is the same as the area of the circle
with diameter 
. Let 
and 
, then simultaneously solve the equations
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
for the sides
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
3. The circles 
and 
inscribed on each half of 
on the arbelos (called Archimedes' circles)
each have diameter 
, or radius 
.
The positions of the circles can be found using the triangles shown above. The lengths of the horizonal legs and hypotenuses are known as indicated, so the vertical legs can be found using the Pythagorean theorem. This then gives the centers of the circles as
 |  |  |
(8)
|
 |  |  |
(9)
|
and
 |  |  |
(10)
|
 |  |  |
(11)
|
4. Let 
be the point at which the circle
centered at 
and of radius 
intersects the enclosing
semicircle, and let 
be the point at which the circle
centered at 
of radius 
intersects the enclosing
semicircle. Then the smallest circle 
passing through 
and tangent to 
is equal to the smallest circle 
passing through 
and tangent to 
(Schoch). Moreover, the radii 
of these circles are the same as Archimedes'
circles. Solving
 |
(12)
|
 |
(13)
|
gives 
, so the center
of 
is
 |  |  |
(14)
|
 |  |  |
(15)
|
Similarly, solving
 |
(16)
|
 |
(17)
|
gives 
, so
the center of 
is
 |  |  |
(18)
|
 |  |  |
(19)
|
5. The Apollonius circle 
of the circles with arcs 
, 
,
and 
is located at a position
 |  |  |
(20)
|
 |  |  |
(21)
|
and has radius 
equal to that of Archimedes' circles (Schoch),
as does the smallest circle 
passing through 
and tangent to 
.
Furthermore, letting 
be the line parallel to 
through the center of circle 
, the circle 
with center on 
and tangent to the small semicircles of the arbelos also
has radius 
(Schoch). The position of the center of 
is given by
 |  |  |
(22)
|
 |  |  |
(23)
|
 |  |  |
(24)
|
The vertical 
position of 
is
 |  |  |
(25)
|
 |  |  |
(26)
|
6. Let 
be the midpoint
of 
, and let 
be the midpoint of 
. Then draw the semicircle
having 
as a diameter
with center 
.
This circle has radius
![R_(PQ)=1/2{1-1/2[r+(1-r)]}=1/4.](/images/equations/Arbelos/NumberedEquation6.svg) |
(27)
|
The smallest circle 
through 
touching arc 
then has radius 
(Schoch). Using similar triangles, the center of this circle is at
 |  |  |
(28)
|
 |  |  |
(29)
|
Similarly, let 
be the point of intersection of 
and the semicircle 
, then the circle
through 
,
, and 
also has radius 
(Schoch). The center of this circle
is at
 |  |  |
(30)
|
 |  |  |
(31)
|
Consider the circle 
of radius 
which is tangent to the two interior semicircles. Its position
and radius are obtained by solving the simultaneous equations
 |
(32)
|
![h^2+(1/2-z)^2=[1/2(1-r)+r_X]^2](/images/equations/Arbelos/NumberedEquation8.svg) |
(33)
|
![(1/2r+r_X)^2+[1/2(1-r)+r_X]^2=(1/4)^2,](/images/equations/Arbelos/NumberedEquation9.svg) |
(34)
|
giving
 |  |  |
(35)
|
 |  |  |
(36)
|
 |  |  |
(37)
|
Letting 
be the smallest circle through 
and tangent to 
, the radius of 
is therefore 
(Schoch), and its center is located at
 |  |  |
(38)
|
 |  |  |
(39)
|
7. Within each small semicircle of an arbelos, construct arbeloses similar to the original. Then the circles 
and 
are congruent and have radius 
(Schoch). Moreover, connect the midpoints
of the arcs and their cusp points to form the rectangles 
and 
. Then these rectangles are similar with respect
to the point 
(Schoch). This point lies on the line 
, and the circle with center 
and radius 
also has radius 
, so 
has coordinates 
.
The following tables summarized the positions of the rectangle vertices.
 | coordinates |  | coordinates |
 |  |  |  |
 |  |  |  |
 |  |  |  |
 |  |  |  |
8. Let 
be the perpendicular
bisector of 
,
let 
be the cusp of the arbelos and 
lie above it, let 
and 
be the tops of the large and small semicircles, respectively. Let 
intersect the lines 
and 
in points 
and 
,
respectively. Then the smallest circle 
passing through 
and tangent to arc 
at 
,
the smallest circle 
through 
and tangent to the outside semicircle at 
, and the circle 
with diameter 
are all Archimedean circles
(Schoch). The circle 
is called the Bankoff circle, and is also the circumcircle of the point 
and tangent points 
and 
of the first Pappus circle. The centers of the circles
, 
, and 
are given by
 |  |  |
(40)
|
 |  |  |
(41)
|
 |  |  |
(42)
|
 |  |  |
(43)
|
 |  |  |
(44)
|
 |  |  |
(45)
|
Rather amazingly, the points 
,
, 
, 
,
, 
, and 
are concyclic (Schoch) in a circle with center 
and radius
 |
(46)
|
9. The smallest circumcircle of Archimedes' circles has an area equal to that of the arbelos.
10. The line tangent to the semicircles 
and 
contains the point 
and 
which lie on the lines 
and 
,
respectively. Furthermore, 
and 
bisect each other, and the points 
, 
, 
,
and 
are concyclic.
11. Construct a chain of tangent circles starting with the circle tangent to
the two small ones and large one. This chain is called a Pappus
chain, and the centers of its circles lie on an ellipse having foci at the centers of the semicircles bounding
it. Furthermore, the diameter of the 
th circle 
is (
)th
the perpendicular distance to the base of the semicircle. This result is most easily proven using inversion, but was known to Pappus, who referred to it
as an ancient theorem (Hood 1961, Cadwell 1966, Gardner 1979, Bankoff 1981).
12. The common tangent 
(see 10) and the common tangent of the great semicircle
and the first Pappus circle meet on line 
.
13. If 
divides 
in the golden ratio 
, then the circles in the chain satisfy
a number of other special properties (Bankoff 1955).
-Aliquot Parts." Forum Geom. 5, 37-45, 2005.
