Cyclic Polygons
The vertices of every triangle are concyclic. (Because of this, some authors only consider sets of four or more points on a circle to be concyclic.) The circle containing the vertices of a triangle is called the circumscribed circle of the triangle. Several other sets of points defined from a triangle are also concyclic, with different circles; see nine-point circle and Lester's theorem.
A quadrilateral ABCD with concyclic vertices is called a cyclic quadrilateral; this happens if and only if (the inscribed angle theorem) which is true if and only if the opposite angles inside the quadrilateral are supplementary.
More generally, a polygon in which all vertices are concyclic is called a cyclic polygon. A polygon is cyclic if and only if the perpendicular bisectors of its edges are concurrent.
Read more about this topic: Concyclic Points