# Arbelos

Given a half circle with diameter AB and any point C on this diameter.

Draw the half circles with diameters AC and BC.

The arbelos is the remaining area (in blue).

The perpendicular in C to AB intersects the circle in D.

We can easily prove that the arbelos area is equal to that of a circle with diameter CD

Hint
Prove that the two circles inscribed in the two parts ACD and BCD of arbelos are equal.
(Archimedes twin circles)

Give a simple construction of these circles.

Details

#### Incircle - Pappus chain

An interresting circle in arbelos is the incircle, tangent to the three circles of arbelos.

A property of this circle :

When the arbelos shape varies (point C varies on AB),
the center I of incircle draws an ellipse.

We may continue inscribing successive circles in arbelos and define
a sequence of incircles : the Pappus chain.

All the centers of these circles lie on an ellipse, for a given arbelos.

At last, choosing a fixed rank k, the locus of the center of the k^{th} circle
in the Pappus chain is an ellipse when C varies.

The case k=1 (The 1st inscribed circle) is then a specific case of a general property
in all circles of the Pappus chain.

Details (applet)

Other properties on incircle : Details (applet)