concyclic

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 $P\ne Q$, then the pencil $𝔓$ 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 $𝔓$.

• 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 $\mathrm{\angle }CAD=\mathrm{\angle }CBD$.