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 where are edge lengths respectively) of the triangle. In terms of the triangle's angles , the barycentric coordinates of the circumcenter are 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 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 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. 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