circumcircle

A circumcircle is a circle that passes through all the vertices of a polygon. For a triangle, the circumcircle is the unique circle through its three vertices and is often called the circumscribed circle. The center of this circle is the circumcenter.

The circumcenter can be constructed as the intersection of the perpendicular bisectors of the triangle's sides. Its position relative to the triangle depends on the angles: it lies inside for acute triangles, at the midpoint of the hypotenuse for right triangles, and outside the triangle for obtuse triangles.

Circumradius and formulas: If a, b, c are the side lengths opposite angles A, B, C, then the circumradius R satisfies R = a/(2 sin A) = b/(2 sin B) = c/(2 sin C). Another common expression is R = abc/(4Δ), where Δ is the area of the triangle. The radius is the distance from the circumcenter to any vertex.

Beyond triangles, a polygon is cyclic if there exists a circle passing through all its vertices; triangles are always cyclic, while not all polygons share this property. The circumcircle plays a central role in many geometric constructions and relationships, including connections to triangle centers and trigonometric calculations.