Schoch line

The Schoch line (cyan) passes through the point A₁.

In geometry, the Schoch line is a line defined from an arbelos and named by Peter Woo after Thomas Schoch, who had studied it in conjunction with the Schoch circles.

Construction

An arbelos is a shape bounded by three mutually-tangent semicircular arcs with collinear endpoints, with the two smaller arcs nested inside the larger one; let the endpoints of these three arcs be (in order along the line containing them) A, B, and C. Let K₁ and K₂ be two more arcs, centered at A and C, respectively, with radii AB and CB, so that these two arcs are tangent at B; let K₃ be the largest of the three arcs of the arbelos. A circle, with the center A₁, is then created tangent to the arcs K₁,K₂, and K₃. This circle is congruent with Archimedes' twin circles, making it an Archimedean circle; it is one of the Schoch circles. The Schoch line is perpendicular to the line AC and passes through the point A₁. It is also the location of the centers of infinitely many Archimedean circles, e.g. the Woo circles.^[1]

Radius and center of A₁

If r = AB/AC, and AC = 1, then the radius of A₁ is

$\rho=\frac{1}{2}r\left(1-r\right)$

and the center is

$\left(r\left(1-r\right)\sqrt{\left(1+r\right)\left(2-r\right)}~,~\frac{1}{2}r\left(1+3r-2r^2\right)\right).$^[1]

References

1. Dodge, Clayton W.; Schoch, Thomas; Woo, Peter Y.; Yiu, Paul (1999), "Those ubiquitous Archimedean circles" (PDF), Mathematics Magazine 72 (3): 202–213, doi:10.2307/2690883, MR 1706441.

Additional reading

• Okumura, Hiroshi; Watanabe, Masayuki (2004), "The Archimedean circles of Schoch and Woo" (PDF), Forum Geometricorum 4: 27–34, MR 2057752.

External links

• van Lamoen, Floor. "Schoch Line." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein". Retrieved 2008-04-11.