EMAT6690

The arbelos is a famous figure believed to have been first studied by Archimedes. It is called the arbelos, from the Greek for "shoemaker's knife," because it resembles the blade of a knife used by ancient cobblers.

It is the yellow shaded region
in the figure below that is bounded by the semicircles with diameters
AB, BC, and AC. B can be any point on AC.

For a GSP sketch that you can drag point B along diameter AC and observe the behavior of the arbelos, click here.

After checking out the GSP sketch and dragging around point B, one should quickly observe that the length of the arc ADC is equal to the sum of the arcs AEB and BFC.

However, no amount of dragging and measuring the arcs will prove the above statement. So, here is a proof of the above conjecture.

Given: arbelos in diagram above

Prove: arclength AEB + arclength BFC = arclength ADC

__PROOF__

Let AO= x, AG= a, thus GO= x-a. And let
BH=b, thus OB= x-2b.

Since C=2pr
and the radius of AO= x, then the arclength of ADC=px, the arclength
of AEB= pa,
and the arclength of AFC= pb.

Now, a= x-a +x-2b, thus 2a=2x-2b.

Therefore, a+b= x.

The arclength of AEB + arclength of AFC= pa + pb.

Factoring p out, we have p (a + b). Making the substitution that a + b
= x, we now have:

arclength AEB + arclength AFC= p x = arclength ADC.

There are some other properties of the arbelos.

1. Draw BD perpendicular to AC. The area
of the arbelos equals the area of a circle with diameter BD.

Click here
for a GSP sketch to investigate this conjecture.

Proof? Click here
for a proof of the investigation above.

2. Construct inside the arbelos what is called
the arbelos train ( which is a train of tangent circles). The
first circle in the train is the broken circle or semicircle with
diameter DB (see the firgure below). It can be continued as far
to left as one would want to go. Conjectures: I. The centers of
all the circles in the train lie on an ellipse; II. The diameter
of any circle C(n) of the train is 1/n th the perpendicular distance
from the center of that circle to the base line AB. This result
was proven by Pappus of Alexandria in the fourth century. It is
sometimes referred as the ancient theorem. The proof of Pappus'
theorem is simple if one uses the principle of inversion geometry.
However, Pappus' did not know inversion methods and so his proof
is more cumbersome. Inversion techniques were not developed until
the 19th century.

For a GSP sketch that you can drag points around click here.

For a GSP script to investigate inversion click here.

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