Schoch line

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.

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.

The Schoch line is perpendicular to the line AC and passes through the point A1.

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