Cyclic quadrilateral: Difference between revisions
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A '''cyclic quadrilateral''' is a [[quadrilateral]] that can be inscribed in a circle. | A '''cyclic quadrilateral''' is a [[quadrilateral]] that can be inscribed in a [[circle]]. While all [[triangles]] are cyclic, the same is not true of quadrilaterals. They have a number of interesting properties. | ||
<center>[[image:Cyclicquad2.png]]</center> | <center>[[image:Cyclicquad2.png]]</center> | ||
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== Properties == | == Properties == | ||
In | In a quadrilateral <math>ABCD</math>: | ||
* <math>\angle A + \angle C = \angle B + \angle D = {180}^{o}</math> | * <math>\angle A + \angle C = \angle B + \angle D = {180}^{o} </math> This property is both sufficient and necessary (Sufficient & necessary = if and only if), and is often used to show that a quadrilateral is cyclic. | ||
* <math>\angle ABD = \angle ACD</math> | * <math>\angle ABD = \angle ACD</math> | ||
* <math>\angle BCA = \angle BDA</math> | * <math>\angle BCA = \angle BDA</math> | ||
* <math>\angle BAC = \angle | * <math>\angle BAC = \angle BDC</math> | ||
* <math>\angle CAD = \angle CBD</math> | * <math>\angle CAD = \angle CBD</math> | ||
* All four [[perpendicular bisector|perpendicular bisectors]] are [[concurrent]]. The converse is also true. This intersection is the [[circumcenter]] of the quadrilateral. | |||
* Any two opposite sites of the quadrilateral are antiparallel with respect to the other two opposite sites. | |||
== Applicable Theorems/Formulae == | == Applicable Theorems/Formulae == | ||
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The following theorems and formulae apply to cyclic quadrilaterals: | The following theorems and formulae apply to cyclic quadrilaterals: | ||
* [[Ptolemy's | * [[Ptolemy's Theorem]] | ||
* [[Brahmagupta's formula]] | * [[Brahmagupta's formula]] | ||
== Problems == | |||
=== Intermediate/Advanced === | |||
* [[1991 AIME Problems/Problem 12]] | |||
* [[2001 AIME I Problems/Problem 13]] | |||
* [[2000 AIME I Problems/Problem 14]] | |||
* [[1997 AIME Problems/Problem 15]] | |||
[[Category:Definition]] | |||
[[Category:Geometry]] | |||
{{stub}} | {{stub}} | ||
Latest revision as of 18:41, 3 January 2025
A cyclic quadrilateral is a quadrilateral that can be inscribed in a circle. While all triangles are cyclic, the same is not true of quadrilaterals. They have a number of interesting properties.
Properties
In a quadrilateral 
:
This property is both sufficient and necessary (Sufficient & necessary = if and only if), and is often used to show that a quadrilateral is cyclic.
- All four perpendicular bisectors are concurrent. The converse is also true. This intersection is the circumcenter of the quadrilateral.
- Any two opposite sites of the quadrilateral are antiparallel with respect to the other two opposite sites.
This property is both sufficient and necessary (Sufficient & necessary = if and only if), and is often used to show that a quadrilateral is cyclic.
Applicable Theorems/Formulae
The following theorems and formulae apply to cyclic quadrilaterals:
Problems
Intermediate/Advanced
- 1991 AIME Problems/Problem 12
- 2001 AIME I Problems/Problem 13
- 2000 AIME I Problems/Problem 14
- 1997 AIME Problems/Problem 15
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