In the arbelos especially in 5th grade math , consider the semicircles K1 and K2 with centers A and C passing through B. The Apollonius circle K3 of K1,  K2 , and the large semicircle of the arbelos is an Archimedean circle A1 . This circle has radius

p=½r(1-r) 

(as it must), and center

A1=(r(1-r)√((1+r)(2-r)), ½r(1+3r-2r²)) 

The line perpendicular to AB  and passing through the center of A1 is called the Schoch line.

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

Let be the radical axis of the great semicircle of the arbelos and K1 . From a point on 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.

References:

Dodge, C. W.; Schoch, T.; Woo, P. Y.; and Yiu, P. “Those Ubiquitous Archimedean Circles.”

Okumura, H. and Watanabe, M. “The Archimedean Circles of Schoch and Woo.”

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