In
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 ...
, the Schoch line is a
line 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
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 ...
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
In geometry, the twin circles are two special circles associated with an arbelos.
An arbelos is determined by three collinear points , , and , and is the curvilinear triangular region between the three semicircles that have , , and as their diame ...
, 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 ca ...
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'', and ''AC'' = 1, then the radius of A
1 is
:
and the center is
:
References
Further reading
*.
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