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 twin circles are two special circles associated with 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 ...
.
An arbelos is determined by three collinear points , , and , and is the curvilinear triangular region between the three
semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line o ...
s that have , , and as their diameters. If the arbelos is partitioned into two smaller regions by a line segment through the middle point of , , and , perpendicular to line , then each of the two twin circles lies within one of these two regions, tangent to its two semicircular sides and to the splitting segment.
These circles first appeared in the ''
Book of Lemmas
The ''Book of Lemmas'' or ''Book of Assumptions'' (Arabic ''Maʾkhūdhāt Mansūba ilā Arshimīdis'') is a book attributed to Archimedes by Thābit ibn Qurra, though the authorship of the book is questionable. It consists of fifteen propositio ...
'', which showed (Proposition V) that the two circles are
congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
.
[
Thābit ibn Qurra, who translated this book into Arabic, attributed it to ]Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
mathematician Archimedes. Based on this claim the twin circles, and several other circles in the Arbelos congruent to them, have also been called Archimedes's circles. However, this attribution has been questioned by later scholarship.
Construction
Specifically, let , , and be the three corners of the arbelos, with between and . Let be the point where the larger semicircle intercepts the line 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 through the point . The segment divides the arbelos in two parts. The twin circles are the two circles inscribed in these parts, each 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 one of the two smaller semicircles, to the segment , and to the largest semicircle.[
Each of the two circles is uniquely determined by its three tangencies. Constructing it is a special case of the ]Problem of Apollonius
In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangencies ...
.
Alternative approaches to constructing two circles congruent to the twin circles have also been found.[ These circles have also been called Archimedean circles. They include the ]Bankoff circle
In geometry, the Bankoff circle or Bankoff triplet circle is a certain Archimedean circle that can be constructed from an arbelos; an Archimedean circle is any circle with area equal to each of Archimedes' twin circles. The Bankoff circle was fi ...
, 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 Americans "Mathematical Games" editor Ma ...
, and 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 tw ...
.
Properties
Let ''a'' and ''b'' be the diameters of two inner semicircles, so that the outer semicircle has diameter ''a'' + ''b''. The diameter of each twin circle is then[
:
Alternatively, if the outer semicircle has unit diameter, and the inner circles have diameters and , the diameter of each twin circle is][
:
The smallest circle that encloses both twin circles has the same area as the arbelos.][
]
See also
* 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.
Construction
An arbelos is a shape bounded by three mutually-tangent semicir ...
References
[{{cite web, author=Weisstein, Eric W, title="Archimedes' Circles." From MathWorld—A Wolfram Web Resource, url=http://mathworld.wolfram.com/ArchimedesCircles.html, accessdate=2008-04-10]
[Thomas Little Heath (1897), ''The Works of Archimedes''. Cambridge University Press. Proposition 5 in the ''Book of Lemmas''. Quote: "''Let AB be the diameter of a semicircle, C any point on AB, and CD perpendicular to it, and let semicircles be described within the first semicircle and having AC, CB as diameters. Then if two circles be drawn touching CD on different sides and each touching two of the semicircles, the circles so drawn will be equal.''" ]
[Floor van Lamoen (2014), ]
A catalog of over fifty Archimedean circles
'' Online document, accessed on 2014-10-08.
[Floor van Lamoen (2014), ]
'' Online document, accessed on 2014-10-08.
Arbelos