to the alternate segment theorem, which 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 and all coordinates are trilinear coordinates: Steiner point: the nonvertex point of intersection of the circumcircle with the Steiner ellipse. (The Steiner ellipse, with center = centroid (), is the ellipse of least area that passes through . An equation for this ellipse is Tarry point: antipode of the Steiner point Focus of the Kiepert parabola: Other properties 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 are the lengths of the sides of the triangle and is the semiperimeter. The expression above is the area of the triangle, by Heron's formula. Trigonometric expressions for the diameter of the circumcircle include 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 and the incenter is where is the incircle radius and is the circumcircle radius; hence the circumradius is at least twice the inradius (Euler's triangle inequality), with equality only in the equilateral