Difference between revisions of "Cyclic quadrilateral"
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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 BDC</math> | * <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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* [[Ptolemy's Theorem]] | * [[Ptolemy's Theorem]] | ||
* [[Brahmagupta's formula]] | * [[Brahmagupta's formula]] | ||
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| + | [[Category:Definition]] | ||
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| + | [[Category:Geometry]] | ||
{{stub}} | {{stub}} | ||
Latest revision as of 19:39, 9 March 2024
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:
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