tangent circles on:
[Wikipedia]
[Google]
[Amazon]
In
Two circles are mutually and externally tangent if distance between their centers is equal to the sum of their radiiWeisstein, Eric W. "Tangent Circles." From MathWorld--A Wolfram Web Resource
/ref>
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 ...
, tangent circles (also known as kissing circles) are circles in a common plane that intersect in a single point. There are two types of tangency
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. More ...
: internal and external. Many problems and constructions in geometry are related to tangent circles; such problems often have real-life applications such as trilateration Trilateration is the use of distances (or "ranges") for determining the unknown position coordinates of a point of interest, often around Earth (geopositioning).
When more than three distances are involved, it may be called multilateration, for emph ...
and maximizing the use of materials.
Two given circles
Two circles are mutually and externally tangent if distance between their centers is equal to the sum of their radii/ref>
Steiner chains
Pappus chains
Three given circles: Apollonius' problem
Apollonius' problem is to construct circles that are tangent to three given circles.Apollonian gasket
If a circle is iteratively inscribed into the interstitial curved triangles between three mutually tangent circles, an Apollonian gasket results, one of the earliest fractals described in print.
Malfatti's problem
Malfatti's problem is to carve three cylinders from a triangular block of marble, using as much of the marble as possible. In 1803,Gian Francesco Malfatti
Giovanni Francesco Giuseppe Malfatti, also known as Gian Francesco or Gianfrancesco (26 September 1731 – 9 October 1807) was an Italian mathematician. He was born in Ala, Trentino, Italy and died in Ferrara.
Malfatti studied at the College of Sa ...
conjectured that the solution would be obtained by inscribing three mutually tangent circles into the triangle (a problem that had previously been considered by Japanese mathematician Ajima Naonobu
, also known as Ajima Manzō Chokuyen, was a Japanese mathematician of the Edo period.Smith, David. (1914).
His Dharma name was (祖眞院智算量空居士).
Work
Ajima is credited with introducing calculus into Japanese mathematics. The si ...
); these circles are now known as the Malfatti circles
In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem o ...
, although the conjecture has been proven to be false.
Six circles theorem
A chain of six circles can be drawn such that each circle is tangent to two sides of a given triangle and also to the preceding circle in the chain. The chain closes; the sixth circle is always tangent to the first circle.Generalizations
Problems involving tangent circles are often generalized to spheres. For example, the Fermat problem of finding sphere(s) tangent to four given spheres is a generalization of Apollonius' problem, whereasSoddy's hexlet
In geometry, Soddy's hexlet is a chain of six spheres (shown in grey in Figure 1), each of which is tangent to both of its neighbors and also to three mutually tangent given spheres. In Figure 1, the three spheres are the red inner sphere and tw ...
is a generalization of a Steiner chain
In geometry, a Steiner chain is a set of circles, all of which are tangent to two given non-intersecting circles (blue and red in Figure 1), where is finite and each circle in the chain is tangent to the previous and next circles in the chain. ...
.
See also
*Tangent lines to circles
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. More ...
* Circle packing theorem
The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles (in gen ...
, the result that every planar graph may be realized by a system of tangent circles
* Hexafoil
The hexafoil is a design with six-fold dihedral symmetry composed from six ''vesica piscis'' lenses arranged radially around a central point, often shown enclosed in a circumference of another six lenses. It is also sometimes known as a "daisy wh ...
, the shape formed by a ring of six tangent circles
* Feuerbach's theorem
In the geometry of triangles, the incircle and nine-point circle of a triangle are internally tangent to each other at the Feuerbach point of the triangle. The Feuerbach point is a triangle center, meaning that its definition does not depend on th ...
on the tangency of the nine-point circle
In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are:
* The midpoint of eac ...
of a triangle with its incircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
and excircles
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
* Descartes' theorem
In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mu ...
* Ford circle
In mathematics, a Ford circle is a circle with center at (p/q,1/(2q^2)) and radius 1/(2q^2), where p/q is an irreducible fraction, i.e. p and q are coprime integers. Each Ford circle is tangent to the horizontal axis y=0, and any two Ford circles ...
* 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 firs ...
* 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 ...
* 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 ''r ...
* 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 ...
* 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 ...
* 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 ...
* Ring lemma
References
External links
* {{MathWorld, title=Tangent circles, urlname=TangentCircles Circles