In any geometry where a circle is defined, a collection of points are said to be concyclic if there is a circle that is incident with all the points.

Remarks. Suppose all points being considered below lie in a Euclidean plane.

• Any two points $P,Q$ are concyclic. In fact, there are infinitely many circles that are incident to both $P$ and $Q$ If $P\neq Q$ then the pencil $\mathfrak{P}$ of circles incident with $P$ and $Q$ share the property that their centers are collinear. It is easy to see that any point on the perpendicular bisector of $\overline{PQ}$ serves as the center of a unique circle in $\mathfrak{P}$
• Any three non-collinear points $P,Q,R$ are concyclic to a unique circle $c$ From the three points, take any two perpendicular bisectors, say of $\overline{PQ}$ and $\overline{PR}$ Then their intersection $O$ is the center of $c$ whose radius is $|OP|$
• Four distinct points $A,B,C,D$ are concyclic iff $\angle CAD=\angle CBD$