Circumcircle

More generally, an n-sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon, or in the special case n = 4, a cyclic quadrilateral.

The circumcenter of a triangle can be constructed by drawing any two of the three perpendicular bisectors.

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.

in the Cartesian plane satisfying the equations guaranteeing that the points A, B, C, v are all the same distance r from the common center

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 similar approach allows one to deduce the equation of the circumsphere of a tetrahedron.

⁠ one parametric equation of the circle starting from the point P0 and proceeding in a positively oriented (i.e., right-handed) sense about ⁠

The isogonal conjugate of the circumcircle is the line at infinity, given in trilinear coordinates by

Additionally, the circumcircle of a triangle embedded in three dimensions can be found using a generalized method.

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, p0, 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: This gives us the following equation for the circumradius r: and the following equation for the circumcenter p0: which can be simplified to: 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

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[2] where α, β, γ are the angles of the triangle.

In terms of the side lengths a, b, c, the trilinears are[3] The circumcenter has barycentric coordinates[4] where a, b, c are edge lengths BC, CA, AB respectively) of the triangle.

In terms of the triangle's angles α, β, γ, the barycentric coordinates of the circumcenter are[3] 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.

As stated previously In Euclidean space, there is a unique circle passing through any given three non-collinear points P1, P2, 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 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: 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.

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: 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

It is common to confuse the minimum bounding circle with the circumcircle.

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.

[7][8] The distance between O and the orthocenter H is[9][10] 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, c is[11] With circumradius R, sides a, b, c, and medians ma, mb, mc, we have[12] If median m, altitude h, and internal bisector t all emanate from the same vertex of a triangle with circumradius R, then[13] 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] A set of points lying on the same circle are called concyclic, and a polygon whose vertices are concyclic is called a cyclic polygon.