
In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, Descartes' theorem states that for every four kissing, or mutually
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
s, the radii of the circles satisfy a certain
quadratic equation
In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after
René Descartes
René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
, who stated it in 1643.
Frederick Soddy
Frederick Soddy FRS (2 September 1877 – 22 September 1956) was an English radiochemist who explained, with Ernest Rutherford, that radioactivity is due to the transmutation of elements, now known to involve nuclear reactions. He also pr ...
's 1936 poem ''The Kiss Precise'' summarizes the theorem in terms of the ''bends'' (signed inverse radii) of the four circles:
Special cases of the theorem apply when one or two of the circles is replaced by a straight line (with zero bend) or when the bends are
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s or
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s. A version of the theorem using
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s allows the centers of the circles, and not just their radii, to be calculated. With an appropriate definition of curvature, the theorem also applies in
spherical geometry
300px, A sphere with a spherical triangle on it.
Spherical geometry or spherics () is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres.
Long studied for its practical applicati ...
and
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
. In higher dimensions, an analogous quadratic equation applies to systems of pairwise tangent spheres or hyperspheres.
History
Geometrical problems involving
tangent circles
In geometry, 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: internal and external. Many problems and constructions in geometry are related to tangen ...
have been pondered for millennia. In ancient Greece of the third century BC,
Apollonius of Perga
Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...
devoted an entire book to the topic,
'Tangencies'' It has been lost, and is known largely through a description of its contents by
Pappus of Alexandria
Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known a ...
and through fragmentary references to it in
medieval Islamic mathematics. However, Greek geometry was largely focused on
straightedge and compass construction
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...
. For instance, the
problem of Apollonius, closely related to Descartes' theorem, asks for the construction of a circle tangent to three given circles which need not themselves be tangent. Instead, Descartes' theorem is formulated using algebraic relations between numbers describing geometric forms. This is characteristic of
analytic geometry
In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineering, and als ...
, a field pioneered by
René Descartes
René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
and
Pierre de Fermat
Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
in the first half of the 17th century.
Descartes discussed the tangent circle problem briefly in 1643, in two letters to Princess
Elisabeth of the Palatinate
Elisabeth of the Palatinate (; 26 December 1618 – 11 February 1680), also known as Elisabeth of Bohemia (), Princess Elisabeth of the Palatinate, or Princess-Abbess of Herford Abbey, was the eldest daughter of Frederick V, Elector Palatine (w ...
. Descartes initially posed to the princess the problem of Apollonius. After Elisabeth's partial results revealed that solving the full problem analytically would be too tedious, he simplified the problem to the case in which the three given circles are mutually tangent, and in solving this simplified problem he came up with the equation describing the relation between the radii, or curvatures, of four pairwise tangent circles. This result became known as Descartes' theorem. Descartes did not provide the reasoning through which he found this relation.
Japanese mathematics
denotes a distinct kind of mathematics which was developed in Japan during the Edo period (1603–1867). The term ''wasan'', from ''wa'' ("Japanese") and ''san'' ("calculation"), was coined in the 1870s and employed to distinguish native Japanese ...
frequently concerned problems involving circles and their tangencies, and Japanese mathematician
Yamaji Nushizumi stated a form of Descartes' circle theorem in 1751. Like Descartes, he expressed it as a polynomial equation on the radii rather than their curvatures. The special case of this theorem for one straight line and three circles was recorded on a Japanese
sangaku
Sangaku or san gaku () are Japanese Euclidean geometry, geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples in Japan, Buddhist temples during the Edo period by members of all so ...
tablet from 1824.
Descartes' theorem was rediscovered in 1826 by
Jakob Steiner
Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry.
Life
Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards st ...
, in 1842 by Philip Beecroft, and in 1936 by
Frederick Soddy
Frederick Soddy FRS (2 September 1877 – 22 September 1956) was an English radiochemist who explained, with Ernest Rutherford, that radioactivity is due to the transmutation of elements, now known to involve nuclear reactions. He also pr ...
. Soddy chose to format his version of the theorem as a poem, ''The Kiss Precise'', and published it in ''
Nature
Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...
''. The kissing circles in this problem are sometimes known as Soddy circles. Soddy also extended the theorem to spheres, and in another poem described the chain of six spheres each tangent to its neighbors and to three given mutually tangent spheres, a configuration now called
Soddy's hexlet.
Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, a ...
and several others extended the theorem and the poem to arbitrary dimensions; Gosset's version was published the following year. The generalization is sometimes called the ''Soddy–Gosset theorem'', although both the hexlet and the three-dimensional version were known earlier, in sangaku and in the 1886 work of Robert Lachlan. A generalization for multiple kissing circles in two dimensions was proved in 2025.
Multiple proofs of the theorem have been published. Steiner's proof uses
Pappus chains and
Viviani's theorem
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from ''any'' interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in ...
. Proofs by Philip Beecroft and by
H. S. M. Coxeter involve four more circles, passing through triples of tangencies of the original three circles; Coxeter also provided a proof using
inversive geometry
In geometry, inversive geometry is the study of ''inversion'', a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry ...
. Additional proofs involve arguments based on symmetry, calculations in
exterior algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
, or algebraic manipulation of
Heron's formula
In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is
A = \sqrt.
It is named after first-century ...
(for which see ). The result also follows from the observation that the
Cayley–Menger determinant
In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a n-dimensional simplex in terms of the squares of all of the distances between pairs of its ...
of the four coplanar circle centers is zero.
Statement
Descartes' theorem is most easily stated in terms of the circles'
curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
s. The ''signed curvature'' (or ''bend'') of a circle is defined where
is its radius. The larger a circle, the smaller is the
magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
of its curvature, and vice versa. The sign in
(represented by the
symbol) is positive for a circle that is ''externally'' tangent to the other circles. For an ''internally'' tangent circle that ''circumscribes'' the other circles, the sign is negative. If a straight line is considered a
degenerate circle with zero curvature (and thus infinite radius), Descartes' theorem also applies to a line and three circles that are all three mutually tangent (see
Generalized circle
In geometry, a generalized circle, sometimes called a ''cline'' or ''circline'',
is a straight line or a circle, the curves of constant curvature in the Euclidean plane.
The natural setting for generalized circles is the extended plane, a plane ...
).
For four circles that are tangent to each other at six distinct points, with curvatures
Descartes' theorem says:
If one of the four curvatures is considered to be a variable, and the rest to be constants, this is a
quadratic equation
In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where the variable (mathematics), variable represents an unknown number, and , , and represent known numbers, where . (If and ...
.
To find the radius of a fourth circle tangent to three given kissing circles, the quadratic equation can be solved as
The
symbol indicates that in general this equation has ''two'' solutions, and any triple of tangent circles has two tangent circles (or degenerate straight lines). Problem-specific criteria may favor one of these two solutions over the other in any given problem.
The theorem does not apply to systems of circles with more than two circles tangent to each other at the same point. It requires that the points of tangency be distinct. When more than two circles are tangent at a single point, there can be infinitely many such circles, with arbitrary curvatures; see
pencil of circles.
Locating the circle centers
To determine a circle completely, not only its radius (or curvature), but also its center must be known. The relevant equation is expressed most clearly if the
Cartesian coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
are interpreted as a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
. The equation then looks similar to Descartes' theorem and is therefore called the ''complex Descartes theorem''. Given four circles with curvatures
and centers
the following equality holds in addition to :
Once
has been found using , one may proceed to calculate
by solving as a quadratic equation, leading to a form similar to :
Again, in general there are two solutions corresponding to the two solutions The plus/minus sign in the above formula does not necessarily correspond to the plus/minus sign in the formula
Special cases
Three congruent circles
When three of the four circles are congruent, their centers form an equilateral triangle, as do their points of tangency. The two possibilities for a fourth circle tangent to all three are concentric, and reduces to
One or more straight lines

If one of the three circles is replaced by
a straight line tangent to the remaining circles, then its curvature is zero and drops out of For instance, then can be factorized
and simplifies
Taking the
square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
of both sides leads to another alternative formulation of this case
which has been described as "a sort of demented version of the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
".
If two circles are replaced by lines, the tangency between the two replaced circles becomes a parallelism between their two replacement lines. In this case, is reduced to the trivial
This corresponds to the observation that, for all four curves to remain mutually tangent, the other two circles must be
Integer curvatures

When four tangent circles described by all have integer curvatures, the alternative fourth circle described by the second solution to the equation must also have an integer curvature. This is because both solutions differ from an integer by the square root of an integer, and so either solution can only be an integer if this square root, and hence the other solution, is also an integer. Every four integers that satisfy the equation in Descartes' theorem form the curvatures of four tangent Integer quadruples of this type are also closely related to
Heronian triangle
In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all positive integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula which Heron demo ...
s, triangles with integer sides and
Starting with any four mutually tangent circles, and repeatedly replacing one of the four with its alternative solution (
Vieta jumping), in all possible ways, leads to a system of infinitely many tangent circles called an
Apollonian gasket. When the initial four circles have integer curvatures, so does each replacement, and therefore all of the circles in the gasket have integer curvatures. Any four tangent circles with integer curvatures belong to exactly one such gasket, uniquely described by its ''root quadruple'' of the largest four largest circles and four smallest curvatures. This quadruple can be found, starting from any other quadruple from the same gasket, by repeatedly replacing the smallest circle by a larger one that solves the same Descartes equation, until no such reduction is possible.
A root quadruple is said to be ''primitive'' if it has no nontrivial
common divisor. Every primitive root quadruple can be found from a factorization of a sum of two squares, as the To be primitive, it must satisfy the additional Factorizations of sums of two squares can be obtained using the
sum of two squares theorem
In number theory, the sum of two squares theorem relates the prime decomposition of any integer to whether it can be written as a sum of two Square number, squares, such that for some integers , .
An integer greater than one can be written as a ...
. Any other integer Apollonian gasket can be formed by multiplying a primitive root quadruple by an arbitrary integer, and any quadruple in one of these gaskets (that is, any integer solution to the Descartes equation) can be formed by reversing the replacement process used to find the root quadruple. For instance, the gasket with root shown in the figure, is generated in this way from the factorized sum of two squares
Ford circles

The special cases of one straight line and integer curvatures combine in the
Ford circles. These are an infinite family of circles tangent to the of the
Cartesian coordinate system
In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
at its rational points. Each
fraction
A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...
(in lowest terms) has a circle tangent to the line at the point
with curvature
. Three of these curvatures, together with the zero curvature of the axis, meet the conditions of Descartes' theorem whenever the denominators of two of the corresponding fractions sum to the denominator of the third. The two Ford circles for fractions
and
(both in lowest terms) are tangent When they are tangent, they form a quadruple of tangent circles with the and with the circle for their
mediant
In music, the mediant (''Latin'': "being in the middle") is the third scale degree () of a diatonic scale, being the note halfway between the tonic and the dominant.Benward & Saker (2003), p.32. In the movable do solfège system, the mediant no ...
.
The Ford circles belong to a special Apollonian gasket with root bounded between two parallel lines, which may be taken as the and the This is the only Apollonian gasket containing a straight line, and not bounded within a negative-curvature circle. The Ford circles are the circles in this gasket that are tangent to the .
Geometric progression

When the four radii of the circles in Descartes' theorem are assumed to be in a
geometric progression
A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the ''common ratio''. For example, the s ...
with the curvatures are also in the same progression (in reverse). Plugging this ratio into the theorem gives the equation
which has only one real solution greater than one, the ratio
where
is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
. If the same progression is continued in both directions, each consecutive four numbers describe circles obeying Descartes' theorem. The resulting double-ended geometric progression of circles can be arranged into a single
spiral pattern of tangent circles, called
Coxeter's loxodromic sequence of tangent circles. It was first described, together with analogous constructions in higher dimensions, by
H. S. M. Coxeter in 1968.
Soddy circles of a triangle
Any
triangle
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
in the plane has three externally tangent circles centered at its vertices. Letting
be the three points,
be the lengths of the opposite sides, and
be the
semiperimeter
In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name ...
, these three circles have radii By Descartes' theorem, two more circles, sometimes called ''Soddy circles'', are tangent to these three circles. They are separated by the
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 ...
, one interior to it and one Descartes' theorem can be used to show that the inner Soddy circle's curvature where
is the triangle's area,
is its
circumradius, and
is its
inradius
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. ...
. The outer Soddy circle has curvature The inner curvature is always positive, but the outer curvature can be positive, negative, or zero. Triangles whose outer circle degenerates to a straight line with curvature zero have been called "Soddyian triangles".
One of the many proofs of Descartes' theorem is based on this connection to triangle geometry and on
Heron's formula
In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, s = \tfrac12(a + b + c), the area is
A = \sqrt.
It is named after first-century ...
for the area of a triangle as a function of its side lengths.
If three circles are externally tangent, with radii
then their centers
form the vertices of a triangle with side lengths
and
and semiperimeter
By Heron's formula, this triangle
has area
Now consider the inner Soddy circle with radius
centered at point
inside the triangle. Triangle
can be broken into three smaller triangles
and
whose areas can be obtained by substituting
for one of the other radii in the area formula above. The area of the first triangle equals the sum of these three areas:
Careful algebraic manipulation shows that this formula is equivalent to , Descartes' theorem.
This analysis covers all cases in which four circles are externally tangent; one is always the inner Soddy circle of the other three. The cases in which one of the circles is internally tangent to the other three and forms their outer Soddy circle are similar. Again the four centers
form four triangles, but (letting
be the center of the outer Soddy circle) the triangle sides incident to
have lengths that are differences of radii,
and
rather than sums.
may lie inside or outside the triangle formed by the other three centers; when it is inside, this triangle's area equals the sum of the other three triangle areas, as above. When it is outside, the quadrilateral formed by the four centers can be subdivided by a diagonal into two triangles, in two different ways, giving an equality between the sum of two triangle areas and the sum of the other two triangle areas. In every case, the area equation reduces to Descartes' theorem. This method does not apply directly to the cases in which one of the circles degenerates to a line, but those can be handled as a limiting case of circles.
Generalizations
Arbitrary four-circle configurations
Descartes' theorem can be expressed as a matrix equation and then generalized to other configurations of four
oriented circle
The Laguerre transformations or axial homographies are an analogue of Möbius transformations over the dual numbers. Originally published as ''Kompleksnye Chisla i Ikh Primenenie v Geometrii'' (in Russian). Moscow: Fizmatgiz. 1963 When studying th ...
s by changing the matrix. Let
be a
column vector
In linear algebra, a column vector with elements is an m \times 1 matrix consisting of a single column of entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some , c ...
of the four circle curvatures and let
be a
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
whose coefficients
represent the relative orientation between the th and th oriented circles at their intersection point:
Then can be rewritten as the
matrix
Matrix (: matrices or matrixes) or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the m ...
equation
As a generalization of Descartes' theorem, a modified symmetric matrix
can represent any desired configuration of four circles by replacing each coefficient with the ''inclination''
between two circles, defined as
where
are the respective radii of the circles, and
is the
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
between their centers. When the circles intersect, the cosine of the intersection angle between the circles. The inclination, sometimes called
inversive distance, is
when the circles are tangent and oriented the same way at their point of tangency,
when the two circles are tangent and oriented oppositely at the point of tangency,
for
orthogonal circles
In geometry, two circles are said to be orthogonal if their respective tangent lines at the points of intersection are perpendicular (meet at a right angle).
A straight line through a circle's center is orthogonal to it, and if straight lines a ...
, outside the interval