circumcircles

A circumcircle is the circle that passes through all vertices of a polygon. For a nondegenerate triangle, there is a unique circumcircle, and its center is called the circumcenter.

In a triangle, the circumcenter is the intersection of the perpendicular bisectors of the sides. The circumcenter is equidistant from all three vertices, and that common distance is the circumradius, denoted R. Hence OA = OB = OC = R for the vertices A, B, and C of the triangle.

The position of the circumcenter depends on the type of triangle. In an acute triangle, the circumcenter lies inside the triangle; in a right triangle, it sits at the midpoint of the hypotenuse; in an obtuse triangle, it lies outside the triangle. The circumcircle is the circle with center O and radius R through all three vertices.

Formulas and relations. If the triangle has side lengths a, b, c opposite angles A, B, C, then R can be expressed as R = a/(2 sin A) = b/(2 sin B) = c/(2 sin C). It is also common to use R = abc/(4Δ), where Δ is the triangle’s area.

Beyond triangles, a circle through the vertices may exist for other polygons; such a polygon is called cyclic. A polygon is cyclic if all its vertices lie on a single circumcircle.

The circumcircle provides a foundational construct in Euclidean geometry, enabling properties and theorems about angles, arcs, and cyclic figures.