Arbelos

Arbelos (shoemaker's knife)

This is the figure ABC of three pairwise tangent half-circles, whose centers are collinear. The picture below shows the whole circles with diameters AB(O₂, r₂), AC(O₁,r₁), CB.
Exercise: To construct an arbelos ABC, filled with a chain of successively tangent circles.
1) Construct a band of successively tangent circles, which are also tangent to two parallels e1, e2, through B and C.
2) Invert this band, with respect to the circle c, with center A and orthogonal to the circle with diameter BC.

The inversion transforms the two lines to the two circles with diameters AB and AC respectively.
The family of equal circles, between the two lines is transformed to the chain of circles filling the arbelos.
Count the inscribed circles c₀, c₁, c₂, c₃, ..., starting with the base c₀ (half circle on BC).
A theorem of Steiner (actually of Pappus, Steiner gave a simpler proof, based on inversion and using this figure) says that the distance d_n of the center of c_n, from the line AB is
d_n = 2*n*r_n, where r_n the radius of c_n.

[Steiner, Werke Bd I, pp. 47-51]

Switch to the [select-on-contour-tool] (CTRL+2). Catch and move point C.
Prove that the centers of circles c_i are on an ellipse f with foci at O₁, O₂ and major axis 2*a = r₁+r₂.
Prove that c, e₁ and the circle with diameter AB intersect at the same points (D and its reflexion on AB).
See the file WooConstruction.html for some elegant ways to construct the first circle c₁.

References

Steiner, J. Werke Bd. I New York, Chelsea, 1971, pp. 47-51.

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Arbelos (shoemaker's knife)

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References