geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the Schoch line is a

line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...

defined from an

arbelos In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that conta ...

and named by Peter Woo after Thomas Schoch, who had studied it in conjunction with the Schoch circles.

Construction

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 ''K''₁ and ''K''₂ 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 ''K''₃ be the largest of the three arcs of the arbelos. A circle, with the center ''A''₁, is then created

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

to the arcs ''K''₁, ''K''₂, and ''K''₃. This circle is congruent with Archimedes' twin circles, making it an

Archimedean circle In geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes' twin circles. If the arbelos is normed such that the diameter of its outer (largest) half circle has a length of 1 and '' ...

; it is one of the Schoch circles. The Schoch line is

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

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 In geometry, the Woo circles, introduced by Peter Y. Woo, are a set of infinitely many Archimedean circles. Construction Form an arbelos with the two inner semicircles tangent at point ''C''. Let ''m'' denote any nonnegative real number In ...

Radius and center of ''A''₁

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

\rho=\fracr\left(1-r\right)

and the center is :

\left(\fracr\left(-1+3r-2r^2\right)~,~r\left(1-r\right)\sqrt\right).

References

External links

*{{cite web, 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 Arbelos de:Archimedischer Kreis#Schoch-Kreise und Schoch-Gerade

Construction

Radius and center of ''A''1

References

Further reading

External links

Radius and center of ''A''₁