Schoch line

In geometry, the Schoch line was created by Thomas Schoch. The line originated from Schoch's dozen circles.

Construction

With an arbelos two circular arcs "K"₁ and "K"₂ are created with the centers at point "A" and "C", respectively. A circle, with the center "A"₁, is then created tangent to the arcs "K"₁,"K"₂, and "K"₃. This circle is congruent with Archimedes' twin circles, making it an Archimedean circle. The Schoch line is perpendicular to the line "AC" and passes through the point "A"₁. It is also the location of the centers of infinitely many Archimedean circles, e.g. the Woo circles.

Radius and center of "A"₁

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

:$ho=frac\{1\}\{2\}rleft(1-r\; ight)$

and the center is

:$left(rleft(1-r\; ight)sqrt\{left(1+r\; ight)left(2-r\; ight)\}~,~frac\{1\}\{2\}rleft(1+3r-2r^2\; ight)\; ight).$

References

*citeweb|author=van Lamoen, Floor|title=Schoch Line." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein|url=http://mathworld.wolfram.com/SchochLine.html|accessdate=2008-04-11