In
geometry, the circumscribed circle or circumcircle of a
polygon is a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
that passes through all the
vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polygon has a circumscribed circle. A polygon that does have one is called a cyclic polygon, or sometimes a concyclic polygon because its vertices are
concyclic. All
triangles, all
regular simple polygons, all
rectangles, all
isosceles trapezoids, and all
right kites are cyclic.
A related notion is the one of a
minimum bounding circle, which is the smallest circle that completely contains the polygon within it, if the circle's center is within the polygon. Every polygon has a unique minimum bounding circle, which may be constructed by a
linear time algorithm. Even if a polygon has a circumscribed circle, it may be different from its minimum bounding circle. For example, for an
obtuse triangle, the minimum bounding circle has the longest side as diameter and does not pass through the opposite vertex.
Triangles
All triangles are cyclic; that is, every triangle has a circumscribed circle.
Straightedge and compass construction
The circumcenter of a triangle can be
constructed by drawing any two of the three
perpendicular bisectors. For three non-collinear points, these two lines cannot be parallel, and the circumcenter is the point where they cross. Any point on the bisector is equidistant from the two points that it bisects, from which it follows that this point, on both bisectors, is equidistant from all three triangle vertices.
The circumradius is the distance from it to any of the three vertices.
Alternative construction
An alternative method to determine the circumcenter is to draw any two lines each one departing from one of the vertices at an angle with the common side, the common angle of departure being 90° minus the angle of the opposite vertex. (In the case of the opposite angle being obtuse, drawing a line at a negative angle means going outside the triangle.)
In
coastal navigation, a triangle's circumcircle is sometimes used as a way of obtaining a
position line using a
sextant
A sextant is a doubly reflecting navigation instrument that measures the angular distance between two visible objects. The primary use of a sextant is to measure the angle between an astronomical object and the horizon for the purposes of cel ...
when no
compass
A compass is a device that shows the cardinal directions used for navigation and geographic orientation. It commonly consists of a magnetized needle or other element, such as a compass card or compass rose, which can pivot to align itself with ...
is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies.
Circumcircle equations
Cartesian coordinates
In the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
, it is possible to give explicitly an equation of the circumcircle in terms of the
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
of the vertices of the inscribed triangle. Suppose that
:
are the coordinates of points . The circumcircle is then the locus of points
in the Cartesian plane satisfying the equations
:
guaranteeing that the points are all the same distance from the common center
of the circle. Using the
polarization identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product the ...
, these equations reduce to the condition that the
matrix
:
has a nonzero
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learni ...
. Thus the circumcircle may alternatively be described as the
locus
Locus (plural loci) is Latin for "place". It may refer to:
Entertainment
* Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front
* ''Locus'' (magazine), science fiction and fantasy magazine
** ''Locus Award' ...
of zeros of the
determinant of this matrix:
:
Using
cofactor expansion, let
:
we then have
where
and – assuming the three points were not in a line (otherwise the circumcircle is that line that can also be seen as a generalized circle with at infinity) –
giving the circumcenter
and the circumradius
A similar approach allows one to deduce the equation of the
circumsphere of a
tetrahedron.
Parametric equation
A
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vec ...
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 can ...
to the plane containing the circle is given by
:
Hence, given the radius, , center, , a point on the circle, and a unit normal of the plane containing the circle, one parametric equation of the circle starting from the point and proceeding in a positively oriented (i.e.,
right-handed
In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more dextrous. The other hand, comparatively often the weaker, less dextrous or simply less subject ...
) sense about is the following:
:
Trilinear and barycentric coordinates
An equation for the circumcircle in
trilinear coordinates is
An equation for the circumcircle in
barycentric coordinates
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
is
The
isogonal conjugate of the circumcircle is the line at infinity, given in
trilinear coordinates by
and in barycentric coordinates by
Higher dimensions
Additionally, the circumcircle of a triangle embedded in dimensions can be found using a generalized method. Let be -dimensional points, which form the vertices of a triangle. We start by transposing the system to place at the origin:
:
The circumradius is then
:
where is the interior angle between and . The circumcenter, , is given by
:
This formula only works in three dimensions as the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
is not defined in other dimensions, but it can be generalized to the other dimensions by replacing the cross products with following identities:
:
Circumcenter coordinates
Cartesian coordinates
The
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
of the circumcenter
are
:
with
:
Without loss of generality this can be expressed in a simplified form after translation of the vertex to the origin of the Cartesian coordinate systems, i.e., when
In this case, the coordinates of the vertices
and
represent the vectors from vertex to these vertices. Observe that this trivial translation is possible for all triangles and the circumcenter
of the triangle follow as
:
with
:
Due to the translation of vertex to the origin, the circumradius can be computed as
:
and the actual circumcenter of follows as
:
Trilinear coordinates
The circumcenter has
trilinear coordinates
:
where are the angles of the triangle.
In terms of the side lengths , the trilinears are
[
:
]
Barycentric coordinates
The circumcenter has barycentric coordinates
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
:
where are edge lengths respectively) of the triangle.
In terms of the triangle's angles , the barycentric coordinates of the circumcenter are[ The circumcenter is listed under X(3).]
:
Circumcenter vector
Since the Cartesian coordinates of any point are a weighted average of those of the vertices, with the weights being the point's barycentric coordinates normalized to sum to unity, the circumcenter vector can be written as
:
Here is the vector of the circumcenter and are the vertex vectors. The divisor here equals where is the area of the triangle. As stated previously
:
Cartesian coordinates from cross- and dot-products
In Euclidean space, there is a unique circle passing through any given three non-collinear points . Using Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
to represent these points as spatial vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors a ...
s, it is possible to use the dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
and cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
to calculate the radius and center of the circle. Let
:
Then the radius of the circle is given by
:
The center of the circle is given by the linear combination
:
where
:
Location relative to the triangle
The circumcenter's position depends on the type of triangle:
*For an acute triangle (all angles smaller than a right angle), the circumcenter always lies inside the triangle.
*For a right triangle, the circumcenter always lies at the midpoint of the hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equ ...
. This is one form of Thales' theorem.
*For an obtuse triangle (a triangle with one angle bigger than a right angle), the circumcenter always lies outside the triangle.
These locational features can be seen by considering the trilinear or barycentric coordinates given above for the circumcenter: all three coordinates are positive for any interior point, at least one coordinate is negative for any exterior point, and one coordinate is zero and two are positive for a non-vertex point on a side of the triangle.
Angles
The angles which the circumscribed circle forms with the sides of the triangle coincide with angles at which sides meet each other. The side opposite angle meets the circle twice: once at each end; in each case at angle (similarly for the other two angles). This is due to the alternate segment theorem, which states that the angle between the tangent and chord equals the angle in the alternate segment.
Triangle centers on the circumcircle of triangle ''ABC''
In this section, the vertex angles are labeled and all coordinates are trilinear coordinates:
* Steiner point: the nonvertex point of intersection of the circumcircle with the Steiner ellipse.
::
:(The Steiner ellipse, with center = centroid (), is the ellipse of least area that passes through . An equation for this ellipse is
*Tarry point
In geometry, the Tarry point for a triangle is a point of concurrency of the lines through the vertices of the triangle perpendicular to the corresponding sides of the triangle's first Brocard triangle . The Tarry point lies on the other end ...
: antipode of the Steiner point
::
*Focus of the Kiepert parabola:
::
Other properties
The diameter of the circumcircle, called the circumdiameter and equal to twice the circumradius, can be computed as the length of any side of the triangle divided by the sine of the opposite angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
:
:
As a consequence of the law of sines, it does not matter which side and opposite angle are taken: the result will be the same.
The diameter of the circumcircle can also be expressed as
:
where are the lengths of the sides of the triangle and is the semiperimeter. The expression above is the area of the triangle, by Heron's formula. Trigonometric expressions for the diameter of the circumcircle include
:
The triangle's nine-point circle has half the diameter of the circumcircle.
In any given triangle, the circumcenter is always collinear with the centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any o ...
and orthocenter. The line that passes through all of them is known as the Euler line
In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, includ ...
.
The isogonal conjugate of the circumcenter is the orthocenter.
The useful minimum bounding circle of three points is defined either by the circumcircle (where three points are on the minimum bounding circle) or by the two points of the longest side of the triangle (where the two points define a diameter of the circle). It is common to confuse the minimum bounding circle with the circumcircle.
The circumcircle of three collinear points
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned ...
is the line on which the three points lie, often referred to as a ''circle of infinite radius''. Nearly collinear points often lead to numerical instability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorit ...
in computation of the circumcircle.
Circumcircles of triangles have an intimate relationship with the Delaunay triangulation
In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle ...
of a set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of points.
By Euler's theorem in geometry
In geometry, Euler's theorem states that the distance ''d'' between the circumcenter and incenter of a triangle is given by
d^2=R (R-2r)
or equivalently
\frac + \frac = \frac,
where R and r denote the circumradius and inradius respectively (the ...
, the distance between the circumcenter and the incenter is
:
where is the incircle radius and is the circumcircle radius; hence the circumradius is at least twice the inradius ( Euler's triangle inequality), with equality only in the equilateral
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
case.[Nelson, Roger, "Euler's triangle inequality via proof without words," ''Mathematics Magazine'' 81(1), February 2008, 58-61.]
The distance between and the orthocenter is
:
For centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any o ...
and nine-point center we have
:
The product of the incircle radius and the circumcircle radius of a triangle with sides is
:
With circumradius , sides , and medians
The Medes (Old Persian: ; Akkadian: , ; Ancient Greek: ; Latin: ) were an ancient Iranian people who spoke the Median language and who inhabited an area known as Media between western and northern Iran. Around the 11th century BC, the ...
, we have
:
If median , altitude , and internal bisector all emanate from the same vertex of a triangle with circumradius , then
:
Carnot's theorem states that the sum of the distances from the circumcenter to the three sides equals the sum of the circumradius and the 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 incen ...
. Here a segment's length is considered to be negative if and only if the segment lies entirely outside the triangle.
If a triangle has two particular circles as its circumcircle and incircle, there exist an infinite number of other triangles with the same circumcircle and incircle, with any point on the circumcircle as a vertex. (This is the case of Poncelet's porism). A necessary and sufficient condition for such triangles to exist is the above equality
Cyclic quadrilaterals
Quadrilaterals that can be circumscribed have particular properties including the fact that opposite angles are supplementary angles
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
(adding up to 180° or π radians).
Cyclic ''n''-gons
For a cyclic polygon with an odd number of sides, all angles are equal if and only if the polygon is regular. A cyclic polygon with an even number of sides has all angles equal if and only if the alternate sides are equal (that is, sides are equal, and sides are equal).
A cyclic pentagon with rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
sides and area is known as a Robbins pentagon
In geometry, a Robbins pentagon is a cyclic pentagon whose side lengths and area are all rational numbers.
History
Robbins pentagons were named by after David P. Robbins, who had previously given a formula for the area of a cyclic pentagon as ...
; in all known cases, its diagonals also have rational lengths.
In any cyclic -gon with even , the sum of one set of alternate angles (the first, third, fifth, etc.) equals the sum of the other set of alternate angles. This can be proven by induction from the case, in each case replacing a side with three more sides and noting that these three new sides together with the old side form a quadrilateral which itself has this property; the alternate angles of the latter quadrilateral represent the additions to the alternate angle sums of the previous -gon.
Let one -gon be inscribed in a circle, and let another -gon be tangential
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 ...
to that circle at the vertices of the first -gon. Then from any point on the circle, the product of the perpendicular distances from to the sides of the first -gon equals the product of the perpendicular distances from to the sides of the second -gon.
Point on the circumcircle
Let a cyclic -gon have vertices on the unit circle. Then for any point on the minor arc , the distances from to the vertices satisfy
:
For a regular -gon, if are the distances from any point on the circumcircle to the vertices , then
:
Polygon circumscribing constant
Any regular polygon is cyclic. Consider a unit circle, then circumscribe a regular triangle such that each side touches the circle. Circumscribe a circle, then circumscribe a square. Again circumscribe a circle, then circumscribe a regular pentagon, and so on. The radii of the circumscribed circles converge to the so-called ''polygon circumscribing constant''
:
. The reciprocal of this constant is the Kepler–Bouwkamp constant.
See also
* Circumcenter of mass
*Circumgon
In mathematics and particularly in elementary geometry, a circumgon is a geometric figure which circumscribes some circle, in the sense that it is the union of the outer edges of non-overlapping triangles each of which has a vertex at the cente ...
*Circumscribed sphere
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle' ...
* Circumcevian triangle
* Inscribed circle
*Japanese theorem for cyclic polygons
__notoc__
In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant.Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929).
Converse ...
*Japanese theorem for cyclic quadrilaterals
In geometry, the Japanese theorem states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle.
Triangulating an arbitrary cyclic quadrilateral by its diagonals yields four overlapping tr ...
*Jung's theorem
In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set. It is named after Heinrich Jung, who first studied this inequality in 1901. Alg ...
, an inequality relating the diameter of a point set to the radius of its minimum bounding sphere
* Kosnita theorem
* Lester's theorem
*Tangential polygon
In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle (also called an ''incircle''). This is a circle that is tangent to each of the polygon's sides. The dual pol ...
* Triangle center
References
External links
Derivation of formula for radius of circumcircle of triangle
at Mathalino.com
at ttp://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches interactive dynamic geometry sketch.
MathWorld
*
*
*{{MathWorld , title=Steiner circumellipse , urlname=SteinerCircumellipse
Interactive
Triangle circumcircle
an
With interactive animation
Circles defined for a triangle
Compass and straightedge constructions