In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

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

Schoch circles

In geometry, the Schoch circles are twelve Archimedean circles constructed by Thomas Schoch.-History:In 1979, Thomas Schoch discovered a dozen new Archimedean circles; he sent his discoveries to Scientific American's "Mathematical Games" editor Martin Gardner. The manuscript was forwarded to Leon...

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Construction

With an arbelos

Arbelos

In geometry, an arbelos is a plane region bounded by a semicircle of diameter 1, connected to semicircles of diameters r and , all oriented the same way and sharing a common baseline. Archimedes is believed to be the first mathematician to study its mathematical properties, as it appears in...

 two circular arcs K₁ and K₂ are created with the centers at point A and C, respectively. A circle, with the center A₁, is then created tangent

Tangent

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

 to the arcs K₁,K₂, and K₃. This circle is congruent with Archimedes' twin circles

Archimedes' circles

In geometry, Archimedes' circles, first created by Archimedes, are two circles that can be created inside of an arbelos, both having the same area as each other.-Construction:...

, making it an Archimedean circle. The Schoch line is perpendicular

Perpendicular

In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

 to the line AC and passes through the point A₁. It is also the location of the centers of infinitely many

Infinity

Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

 Archimedean circles, e.g. the Woo circles

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. Draw two circles, with radii m times the radius of the smaller two...

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Radius and center of A₁

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


and the center is