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.

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 kth 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)


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