[alogo] 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 dn of the center of cn, from the line AB is
dn = 2*n*rn, where rn the radius of cn.

[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 ci 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.

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]
[2_0] [2_1] [2_2]

See Also



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

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