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| − | {{stub}}
| + | '''Cyclic''' generally means to go in cycles. |
| − | | + | * [[Cyclic polygon]] |
| − | A [[polygon]] is '''cyclic''' if it can be [[inscribe]]d in a [[circle]], that is, if there exists a circle so that every [[vertex]] of the polygon lies on the circle. All ([[nondegenerate]]) [[triangle]]s and all [[regular polygon]]s are cyclic. When talking about a cyclic polygon, the circle in which it can be inscribed is called its [[circumcircle]]. The [[radius]] of this circle is known as the [[circumradius]] of the polygon.
| + | * [[Cyclic group]] |
| − | | + | {{disambig}} |
| − | Because two different circles intersect in at most two points, any polygon can be inscribed in at most one circle.
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| − | Since all nondegenerate triangles are cyclic, the simplest polygon for which it is interesting to consider cyclicity is the [[quadrilateral]]. The existence of cyclic quadrilaterals in a geometry problem often suggests [[angle chasing]]. Given a nondegenerate, [[convex]] quadrilateral <math>ABCD</math> (with vertices in that order) in the plane, the following conditions are all [[equivalent]]:
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| − | * <math>ABCD</math> (with vertices in that order) is cyclic
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| − | * <math>\angle ACB</math> and <math>\angle ADB</math> are equal (this also holds for three other pairs of angles, found by considering equivalent quadrilaterals <math>BCDA</math>, <math>CDAB</math>, <math>DABC</math>)
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| − | * <math>\angle ABC</math> and <math>\angle CDA</math> are [[supplementary]] (this also holds for the other pair of angles <math>\angle BCD</math> and <math>\angle DAB</math>. | |
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| − | The above approach requires you to be able to determine the order of vertices of the cyclic quadrilateral. Sometimes, given a problem, there is more than one possible order for some cyclic quadrilateral, and the problem of [[configuration dependence]] arises. Often, this problem can be circumvented through the usage of [[directed angles]] (but directed angles have their own pitfalls, so be careful).
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| − | ==See also==
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| − | * [[Circumscribe]] | |
| − | * [[Circumradius]]
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| − | * [[Polygon]]
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| − | [[Category:Geometry]]
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Latest revision as of 12:57, 11 April 2025
Cyclic generally means to go in cycles.
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