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≠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 ¯PQ serves as the center of a unique circle in 𝔓.
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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 ¯PQ and ¯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 ∠CAD=∠CBD.
Title | concyclic |
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Canonical name | Concyclic |
Date of creation | 2013-03-22 16:07:58 |
Last modified on | 2013-03-22 16:07:58 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 7 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 51-00 |