In geometry, the **circumscribed circle** or **circumcircle** of a polygon is a circle 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**.

- Triangles
- Straightedge and compass construction
- Alternative construction
- Circumcircle equations
- Circumcenter coordinates
- Angles
- Triangle centers on the circumcircle of triangle ABC
- Other properties
- Cyclic quadrilaterals
- Cyclic n-gons
- Point on the circumcircle
- Polygon circumscribing constant
- See also
- References
- External links
- MathWorld
- Interactive

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.^{ [1] } 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.

All triangles are cyclic; that is, every triangle has a circumscribed circle.

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.

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 when no compass is available. The horizontal angle between two landmarks defines the circumcircle upon which the observer lies.

In the Euclidean plane, it is possible to give explicitly an equation of the circumcircle in terms of the Cartesian coordinates of the vertices of the inscribed triangle. Suppose that

are the coordinates of points *A*, *B*, and *C*. The circumcircle is then the locus of points **v** = (*v*_{x},*v*_{y}) in the Cartesian plane satisfying the equations

guaranteeing that the points **A**, **B**, **C**, and **v** are all the same distance *r* from the common center *u* of the circle. Using the polarization identity, these equations reduce to the condition that the matrix

has a nonzero kernel. Thus the circumcircle may alternatively be described as the locus of zeros of the determinant of this matrix:

Using cofactor expansion, let

we then have a|**v**|^{2} − 2**Sv** − *b* = 0 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 S at infinity), |**v** − **S**/*a*|^{2} = *b*/*a* + |**S**|^{2}/*a*^{2}, giving the circumcenter **S**/*a* and the circumradius √*b*/*a* + |**S**|^{2}/*a*^{2}. A similar approach allows one to deduce the equation of the circumsphere of a tetrahedron.

A unit vector perpendicular to the plane containing the circle is given by

Hence, given the radius, *r*, center, *P _{c}*, a point on the circle,

An equation for the circumcircle in trilinear coordinates *x* : *y* : *z* is^{ [2] }*a*/*x* + *b*/*y* + *c*/*z* = 0. An equation for the circumcircle in barycentric coordinates *x* : *y* : *z* is *a*^{2}/*x* + *b*^{2}/*y* + *c*^{2}/*z* = 0.

The isogonal conjugate of the circumcircle is the line at infinity, given in trilinear coordinates by *ax* + *by* + *cz* = 0 and in barycentric coordinates by *x* + *y* + *z* = 0.

Additionally, the circumcircle of a triangle embedded in *d* dimensions can be found using a generalized method. Let **A**, **B**, and **C** be *d*-dimensional points, which form the vertices of a triangle. We start by transposing the system to place **C** at the origin:

The circumradius, *r*, is then

where *θ* is the interior angle between **a** and **b**. The circumcenter, *p*_{0}, is given by

This formula only works in three dimensions as the cross product is not defined in other dimensions, but it can be generalized to the other dimensions by replacing the cross products with following identities:

The Cartesian coordinates of the circumcenter are

with

Without loss of generality this can be expressed in a simplified form after translation of the vertex *A* to the origin of the Cartesian coordinate systems, i.e., when *A*′ = *A* − *A* = (*A*′_{x},*A*′_{y}) = (0,0). In this case, the coordinates of the vertices *B*′ = *B* − *A* and *C*′ = *C* − *A* represent the vectors from vertex *A*′ to these vertices. Observe that this trivial translation is possible for all triangles and the circumcenter of the triangle *A*′*B*′*C*′ follow as

with

Due to the translation of vertex *A* to the origin, the circumradius *r* can be computed as

and the actual circumcenter of *ABC* follows as

The circumcenter has trilinear coordinates ^{ [3] }

- cos
*α*: cos*β*: cos*γ*

where *α*, *β*, *γ* are the angles of the triangle.

In terms of the side lengths *a, b, c*, the trilinears are^{ [4] }

The circumcenter has barycentric coordinates

^{ [5] }

where *a*, *b*, *c* are edge lengths (*BC*, *CA*, *AB* respectively) of the triangle.

In terms of the triangle's angles the barycentric coordinates of the circumcenter are^{ [4] }

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 *U* is the vector of the circumcenter and * A, B, C * are the vertex vectors. The divisor here equals 16*S*^{2} where *S* is the area of the triangle. As stated previously

In Euclidean space, there is a unique circle passing through any given three non-collinear points *P*_{1}, *P*_{2}, and *P*_{3}. Using Cartesian coordinates to represent these points as spatial vectors, it is possible to use the dot product and cross product 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

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. 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.

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 *α* meets the circle twice: once at each end; in each case at angle *α* (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.

In this section, the vertex angles are labeled *A*, *B*, *C* and all coordinates are trilinear coordinates:

- Steiner point =
*bc*/ (*b*^{2}−*c*^{2}) :*ca*/ (*c*^{2}−*a*^{2}) :*ab*/ (*a*^{2}−*b*^{2}) = the nonvertex point of intersection of the circumcircle with the Steiner ellipse. (The Steiner ellipse, with center = centroid (*ABC*), is the ellipse of least area that passes through*A*,*B*, and*C*. An equation for this ellipse is 1/(*ax*) + 1/(*by*) + 1/(*cz*) = 0.) - Tarry point = sec (
*A*+ ω) : sec (*B*+ ω) : sec (*C*+ ω) = antipode of the Steiner point - Focus of the Kiepert parabola = csc (
*B*−*C*) : csc (*C*−*A*) : csc (*A*−*B*).

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:

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 *a*, *b*, *c* are the lengths of the sides of the triangle and *s* = (*a* + *b* + *c*)/2 is the semiperimeter. The expression above is the area of the triangle, by Heron's formula.^{ [6] } Trigonometric expressions for the diameter of the circumcircle include^{ [7] }

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 and orthocenter. The line that passes through all of them is known as the Euler line.

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 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 computation of the circumcircle.

Circumcircles of triangles have an intimate relationship with the Delaunay triangulation of a set of points.

By Euler's theorem in geometry, the distance between the circumcenter *O* and the incenter *I* is

where *r* is the incircle radius and *R* is the circumcircle radius; hence the circumradius is at least twice the inradius (Euler's triangle inequality), with equality only in the equilateral case.^{ [8] }^{ [9] }

The distance between *O* and the orthocenter *H* is^{ [10] }^{ [11] }

For centroid *G* and nine-point center *N* we have

The product of the incircle radius and the circumcircle radius of a triangle with sides *a*, *b*, and *c* is^{ [12] }

With circumradius *R*, sides *a*, *b*, *c*, and medians *m*_{a}, *m*_{b}, and *m*_{c}, we have^{ [13] }

If median *m*, altitude *h*, and internal bisector *t* all emanate from the same vertex of a triangle with circumradius *R*, then^{ [14] }

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.^{ [15] } 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 *n* = 3 case of Poncelet's porism). A necessary and sufficient condition for such triangles to exist is the above equality ^{ [16] }

Quadrilaterals that can be circumscribed have particular properties including the fact that opposite angles are supplementary angles (adding up to 180° or π radians).

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 1, 3, 5, ... are equal, and sides 2, 4, 6, ... are equal).^{ [17] }

A cyclic pentagon with rational sides and area is known as a Robbins pentagon; in all known cases, its diagonals also have rational lengths.^{ [18] }

In any cyclic *n*-gon with even *n*, 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 *n*=4 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 *n*-gon.

Let one *n*-gon be inscribed in a circle, and let another *n*-gon be tangential to that circle at the vertices of the first *n*-gon. Then from any point *P* on the circle, the product of the perpendicular distances from *P* to the sides of the first *n*-gon equals the product of the perpendicular distances from *P* to the sides of the second *n*-gon.^{ [19] }

Let a cyclic *n*-gon have vertices *A*_{1} , ..., *A*_{n} on the unit circle. Then for any point *M* on the minor arc *A*_{1}*A*_{n}, the distances from *M* to the vertices satisfy^{ [20] }

For a regular *n*-gon, if are the distances from any point on the circumcircle to the vertices , then ^{ [21] }

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*

(sequence A051762 in the OEIS ). The reciprocal of this constant is the Kepler–Bouwkamp constant.

- Circumcenter of mass
- Circumgon
- Circumscribed sphere
- Inscribed circle
- Japanese theorem for cyclic polygons
- Japanese theorem for cyclic quadrilaterals
- Jung's theorem, an inequality relating the diameter of a point set to the radius of its minimum bounding sphere
- Kosnita theorem
- Lester's theorem
- Tangential polygon
- Triangle center

In geometry, a **tetrahedron**, also known as a **triangular pyramid**, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces.

A **triangle** is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices *A*, *B*, and *C* is denoted .

**Angular displacement** of a body is the angle in radians, degrees or revolutions through which a point revolves around a centre or a specified axis in a specified sense. When a body rotates about its axis, the motion cannot simply be analyzed as a particle, as in circular motion it undergoes a changing velocity and acceleration at any time (*t*). When dealing with the rotation of a body, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the body's motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal and negligible. Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion.

In trigonometry, the **law of sines**, **sine law**, **sine formula**, or **sine rule** is an equation relating the lengths of the sides of a triangle to the sines of its angles. According to the law,

An **ellipsoid** is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In geometry, an **altitude** of a triangle is a line segment through a vertex and perpendicular to a line containing the base. This line containing the opposite side is called the *extended base* of the altitude. The intersection of the extended base and the altitude is called the *foot* of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. The process of drawing the altitude from the vertex to the foot is known as *dropping the altitude* at that vertex. It is a special case of orthogonal projection.

In geometry, the **incircle** or **inscribed circle** of a triangle is the largest circle contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.

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 other and are each 60°. It is also a regular polygon, so it is also referred to as a **regular triangle**.

In mechanics and geometry, the **3D rotation group**, often denoted **SO(3)**, is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.

In Euclidean geometry, a **cyclic quadrilateral** or **inscribed quadrilateral** is a quadrilateral whose vertices all lie on a single circle. This circle is called the *circumcircle* or circumscribed circle, and the vertices are said to be *concyclic*. The center of the circle and its radius are called the *circumcenter* and the *circumradius* respectively. Other names for these quadrilaterals are **concyclic quadrilateral** and **chordal quadrilateral**, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

In the mathematical field of differential geometry, one definition of a **metric tensor** is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar *g*(*v*, *w*) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.

In geometry, **Thales's theorem** states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's *Elements*. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.

In geometry, the **incenter** of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.

In geometry, a set of points are said to be **concyclic** if they lie on a common circle. All concyclic points are at the same distance from the center of the circle. Three points in the plane that do not all fall on a straight line are concyclic, but four or more such points in the plane are not necessarily concyclic.

In linear algebra, a **rotation matrix** is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix

In geometry, **Euler's rotation theorem** states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a *rotation group*.

In geometry, the area enclosed by a circle of radius r is π*r*^{2}. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.1416.

In geometry, various **formalisms** exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

In geometry, a **bicentric polygon** is a tangential polygon which is also cyclic — that is, inscribed in an outer circle that passes through each vertex of the polygon. All triangles and all regular polygons are bicentric. On the other hand, a rectangle with unequal sides is not bicentric, because no circle can be tangent to all four sides.

In Euclidean geometry, a **bicentric quadrilateral** is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called *inradius* and *circumradius*, and *incenter* and *circumcenter* respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are **chord-tangent quadrilateral** and **inscribed and circumscribed quadrilateral**. It has also rarely been called a *double circle quadrilateral* and *double scribed quadrilateral*.

- ↑ Megiddo, N. (1983). "Linear-time algorithms for linear programming in
**R**^{3}and related problems".*SIAM Journal on Computing*.**12**(4): 759–776. doi:10.1137/0212052. - ↑ Whitworth, William Allen (1866).
*Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimensions*. Deighton, Bell, and Co. p. 199. - ↑ Whitworth (1866), p. 19.
- 1 2 Kimberling, Clark. "Part I: Introduction and Centers X(1) – X(1000)".
*Encyclopedia of Triangle Centers*. The circumcenter is listed under X(3). - ↑ Weisstein, Eric W. "Barycentric Coordinates".
*MathWorld*. - ↑ Coxeter, H.S.M. (1969). "Chapter 1".
*Introduction to geometry*. Wiley. pp. 12–13. ISBN 0-471-50458-0. - ↑ Dörrie, Heinrich (1965).
*100 Great Problems of Elementary Mathematics*. Dover. p. 379. - ↑ Nelson, Roger, "Euler's triangle inequality via proof without words,"
*Mathematics Magazine*81(1), February 2008, 58-61. - ↑ Svrtan, Dragutin; Veljan, Darko (2012). "Non-Euclidean versions of some classical triangle inequalities".
*Forum Geometricorum*.**12**: 197–209. See in particular p. 198. - ↑ Gras, Marie-Nicole (2014). "Distances between the circumcenter of the extouch triangle and the classical centers".
*Forum Geometricorum*.**14**: 51–61. - ↑ Smith, G. C.; Leversha, Gerry (November 2007). "Euler and triangle geometry".
*The Mathematical Gazette*.**91**(522): 436–452. JSTOR 40378417. See in particular p. 449. - ↑ Johnson, Roger A. (1929).
*Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle*. Houghton Mifflin Co. p. 189, #298(d). hdl:2027/wu.89043163211. Republished by Dover Publications as*Advanced Euclidean Geometry*, 1960 and 2007. - ↑ Posamentier, Alfred S.; Lehmann, Ingmar (2012).
*The Secrets of Triangles*. Prometheus Books. pp. 289–290. - ↑ Altshiller Court, Nathan (1952).
*College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle*(2nd ed.). Barnes & Noble. p. 122, #96. Reprinted by Dover Publications, 2007. - ↑ Altshiller Court (1952), p. 83.
- ↑ Johnson (1929), p. 188.
- ↑ De Villiers, Michael (March 2011). "95.14 Equiangular cyclic and equilateral circumscribed polygons".
*The Mathematical Gazette*.**95**(532): 102–107. JSTOR 23248632. - ↑ Buchholz, Ralph H.; MacDougall, James A. (2008). "Cyclic polygons with rational sides and area".
*Journal of Number Theory*.**128**(1): 17–48. doi: 10.1016/j.jnt.2007.05.005 . MR 2382768. Archived from the original on 2018-11-12. - ↑ Johnson (1929), p. 72.
- ↑ "Inequalities proposed in
*Crux Mathematicorum*" (PDF).*The IMO Compendium*. p. 190, #332.10. - ↑ Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids".
*Communications in Mathematics and Applications*.**11**: 335–355.

- Derivation of formula for radius of circumcircle of triangle at Mathalino.com
- Semi-regular angle-gons and side-gons: respective generalizations of rectangles and rhombi at Dynamic Geometry Sketches, interactive dynamic geometry sketch.

- Triangle circumcircle and circumcenter With interactive animation
- An interactive Java applet for the circumcenter

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