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, 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 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 perpendicular 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) 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 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 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 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