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 ....

, 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

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...

: 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

In geometry, trilateration is the process of determinating absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. In addition to its interest as a geometric problem, trilateration does have practical applications in surveying and...

 and maximizing the use of materials.

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 interstital 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

Gian Francesco Malfatti

Giovanni Francesco Giuseppe, also known as Gian Francesco or Gianfrancesco was an Italian mathematician. He was born in Ala, Trentino, Italy and died in Ferrara....

 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); these circles are now known as the Malfatti circles

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...

, 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.

Seven circles theorem

A chain of six tangent circles is given, each of which is tangent to a seventh given circle. The tangent points of the chain circles with the seventh circle are connected pairwise between opposite circles in the chain, i.e., between circles 1 and 4, 2 and 5 and 3 and 6. These lies are concurrent, i.e., they intersect in the same point.

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

Problem of Apollonius

In Euclidean plane geometry, Apollonius' problem is to construct circles that are tangent to three given circles in a plane . Apollonius of Perga posed and solved this famous problem in his work ; this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived...

, whereas Soddy's hexlet

Soddy's hexlet

In geometry, Soddy's hexlet is a chain of six spheres , each of which is tangent to both of its neighbors and also to three mutually tangent given spheres. In Figure 1, these three spheres are shown as an outer circumscribing sphere C , and two spheres A and B above and below the plane of their...

 is a generalization of a Steiner chain

Steiner chain

In geometry, a Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles , where n 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
  Tangent lines to circles
  In Euclidean plane geometry, tangent lines to circles form the subject of several theorems, and play an important role in many geometrical constructions and proofs...
  
  
• Circle packing theorem
  Circle packing theorem
  The circle packing theorem describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles whose interiors are disjoint...
  
  , the result that every planar graph may be realized by a system of tangent circles