Schoch Line


In the arbelos, consider the semicircles K_1 and K_2 with centers A and C passing through B. The Apollonius circle K_3 of K_1, K_2 and the large semicircle of the arbelos is an Archimedean circle A_1. This circle has radius


(as it must), and center


The line perpendicular to AB and passing through the center of A_1 is called the Schoch line.

Now let K_a and K_c be two semicircles through C with radii proportional to AC and BC respectively. The circle tangent to K_a and K_c with its center on the Schoch line is an Archimedean circle. These circles are called Woo circles.

Let l_1 be the radical axis of the great semicircle of the arbelos and K_1. From a point on l_1 consider the tangents to the circle on diameter BC. The circle with center on the Schoch line and tangent to these tangents is a Woo circle (Okumura and Watanabe 2004).

An applet for investigating Woo circles and Schoch lines has been prepared by Schoch (2005).

See also

Arbelos, Archimedes' Circles, Woo Circle

This entry contributed by Floor van Lamoen

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Dodge, C. W.; Schoch, T.; Woo, P. Y.; and Yiu, P. "Those Ubiquitous Archimedean Circles." Math. Mag. 72, 202-213, 1999.Okumura, H. and Watanabe, M. "The Archimedean Circles of Schoch and Woo." Forum Geom. 4, 27-34, 2004., T. "Arbelos: The Woo Circles." 2005.

Referenced on Wolfram|Alpha

Schoch Line

Cite this as:

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

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