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^{}.

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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 $\U0001d513$ 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 $\U0001d513$.

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Any three noncollinear 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 $OP$.

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Four distinct points $A,B,C,D$ are concyclic iff $\mathrm{\angle}CAD=\mathrm{\angle}CBD$.
Title  concyclic 

Canonical name  Concyclic 
Date of creation  20130322 16:07:58 
Last modified on  20130322 16:07:58 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 5100 