# Extensive Definition

In geometry, the circumscribed
circle or circumcircle of a polygon is a circle which passes
through all the vertices of the polygon. The center
of this circle is called the circumcenter.

A polygon which has a circumscribed circle is
called a cyclic polygon. All regular
simple
polygons, all triangles
and all rectangles 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. Not every polygon has a circumscribed circle, as
the vertices of a polygon do not need to all lie on a circle, but
every polygon has unique minimum bounding circle, which may be
constructed by a linear time
algorithm.ref Megiddo Even if a
polygon has a circumscribed circle, it may not coincide with 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.

## Circumcircles of triangles

All triangles are cyclic, i.e. every triangle has
a circumscribed circle.

The circumcenter of a triangle can be found as
the intersection of the three perpendicular bisectors. (A perpendicular
bisector is a line that forms a right angle with one of the
triangle's sides and intersects that side at its midpoint.) This is
because the circumcenter is equidistant from any pair of the
triangle's points, and all points on the perpendicular bisectors
are equidistant from those points of 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.

The circumcenter's position depends on the type
of triangle:

- If and only if a triangle is acute (all angles smaller than a right angle), the circumcenter lies inside the triangle
- If and only if it is obtuse (has one angle bigger than a right angle), the circumcenter lies outside
- If and only if it is a right triangle, the circumcenter lies on one of its sides (namely, the hypotenuse). This is one form of Thales' theorem.

The diameter of the circumcircle
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 doesn't matter which side is taken: the result will
be the same.) The triangle's nine-point
circle has half the diameter of the circumcircle. The diameter
of the circumcircle of the triangle ΔABC is

\begin \text & = \frac\\ & = \frac\\
& = \frac \end

where a, b, c are the lengths of the sides of the
triangle and
s = (a + b + c)/2
is the semiperimeter. The radical in the second denominator above
is the area of the triangle, by Heron's
formula.ref Coxeter

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.

### Circumcircle equations

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. Thus
suppose that

- \mathbf = (A_x,A_y)
- \mathbf = (B_x,B_y)
- \mathbf = (C_x,C_y)

are the coordinates of points A, B, and C. The
circumcircle is then the locus of points v = (vx,vy) in the
Cartesian plane satisfying the equations

- |\mathbf-\mathbf|^2 - r^2 = 0
- |\mathbf-\mathbf|^2 - r^2 = 0
- |\mathbf-\mathbf|^2 - r^2 = 0
- |\mathbf-\mathbf|^2 - r^2 = 0

guaranteeing that the points A, B, v are all the
same distance r2 from the common center u of the circle. Using the
polarization
identity, these equations reduce to a the condition that the
matrix

- \begin

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

- \det\begin

Expanding by cofactor
expansion, let

- \quad

- a=\det\begin

An equation for the circumcircle in trilinear
coordinates x : y : z is a/x + b/y + c/z = 0. An equation for
the circumcircle in barycentric
coordinates x : y : z is 1/x + 1/y + 1/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.

### Coordinates of circumcenter

The circumcenter has trilinear
coordinates (cos \alpha, cos \beta, cos \gamma) where \alpha,
\beta, \gamma are the angles of the triangle. The circumcenter has
barycentric coordinates

- \left(a^2(-a^2+b^2+c^2),\;b^2(a^2-b^2+c^2),\;c^2(a^2+b^2-c^2)\right),

where a,b,c are edge lengths (BC,CA,AB
respectively) of the triangle. The Cartesian
coordinates are discussed below.

### Using the cross and dot product

In Euclidean
space, there is a unique circle passing through any given three
non-collinear points P1, P2, and P3. 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

\mathrm = \begin x_1 \\ y_1 \\ z_1 \end, \mathrm
= \begin x_2 \\ y_2 \\ z_2 \end, \mathrm = \begin x_3 \\ y_3 \\ z_3
\end

Then the radius of the circle is given by

\mathrm = \frac The center of the circle is given
by the linear
combination

\mathrm = \alpha \, P_1 + \beta \, P_2 + \gamma
\, P_3

where

\alpha = \frac \beta = \frac \gamma = \frac

#### Parametric equation

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

\hat = \frac

Hence, given the radius, \mathrm , center,
\mathrm, a point on the circle, \mathrm and a unit normal of the
plane containing the circle, \hat, one parametric equation of the
circle starting from the point \mathrm and proceeding in a
positively oriented (i.e., right-handed)
sense about \hat is the following:

\mathrm \left( s \right) = \mathrm + \cos \left(
\frac \right) \left( P_0 - P_c \right) + \sin \left( \frac \right)
\left[ \hat \times \left( P_0 - P_c \right) \right]

### The angles at which the circle meets the sides

The angles at which the circumscribed circle meet
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). The alternate segment theorem
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 A,
B, C and all coordinates are trilinear
coordinates:

- Steiner point = bc/ (b2 − c2) : ca/ (c2 − a2) : ab/(a2 − b2) = 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)

## Cyclic quadrilaterals

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

- inscribed circle
- Jung's theorem, an inequality relating the diameter of a point set to the radius of its minimum bounding circle
- Lester's theorem
- Circumscribed sphere

## References

- note CoxeterIntroduction to geometry

- note Megiddo

- note PedoeGeometry: a comprehensive course

## External links

- Circumscribed Circle with Known Coordinates of Vertices of a Triangle at Geometry Atlas.
- Triangle centers by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
- Triangle circumcircle and circumcenter With interactive animation
- Circumcircle at MathWorld
- Steiner circumellipse at MathWorld
- An interactive Java applet for the circumcenter

circumcircle in Asturian: Circuncentru

circumcircle in Catalan: Circumcentre

circumcircle in Czech: Kružnice opsaná

circumcircle in German: Umkreis

circumcircle in Esperanto: Ĉirkaŭskribita
cirklo

circumcircle in Spanish: Circuncentro

circumcircle in French: Cercle circonscrit

circumcircle in Galician: Circuncentro

circumcircle in Korean: 외접원

circumcircle in Italian: Circumcerchio

circumcircle in Hungarian: Köréírt kör

circumcircle in Dutch: Omgeschreven cirkel

circumcircle in Japanese: 外接円

circumcircle in Low German: Ümkrink

circumcircle in Polish: Okrąg opisany na
wielokącie

circumcircle in Swedish: Omskriven cirkel

circumcircle in Tamil: சூழ்தொடு வட்டம்

circumcircle in Chinese: 外接圓