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A polygon is cyclic if it can be inscribed in a circle, that is, if there exists a circle so that every vertex of the polygon lies on the circle. All (nondegenerate) triangles and all regular polygons are cyclic. When talking about a cyclic polygon, the circle in which it can be inscribed is called its circumcircle. The radius of this circle is known as the circumradius of the polygon.

Because two different circles intersect in at most two points, any polygon can be inscribed in at most one circle.

Since all nondegenerate triangles are cyclic, the simplest polygon for which it is interesting to consider cyclicity is the quadrilateral. The existence of cyclic quadrilaterals in a geometry problem often suggests angle chasing. Given a nondegenerate, convex quadrilateral $ABCD$ (with vertices in that order) in the plane, the following conditions are all equivalent:

  • $ABCD$ (with vertices in that order) is cyclic
  • $\angle ACB$ and $\angle ADB$ are equal (this also holds for three other pairs of angles, found by considering equivalent quadrilaterals $BCDA$, $CDAB$, $DABC$)
  • $\angle ABC$ and $\angle CDA$ are supplementary (this also holds for the other pair of angles $\angle BCD$ and $\angle DAB$.

The above approach requires you to be able to determine the order of vertices of the cyclic quadrilateral. Sometimes, given a problem, there is more than one possible order for some cyclic quadrilateral, and the problem of configuration dependence arises. Often, this problem can be circumvented through the usage of directed angles (but directed angles have their own pitfalls, so be careful).

See also