Difference between revisions of "Cyclic quadrilateral"
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* All four [[perpendicular bisector|perpendicular bisectors]] are [[concurrent]]. The converse is also true. This intersection is the [[circumcenter]] of the quadrilateral. | * 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. | * Any two opposite sites of the quadrilateral are antiparallel with respect to the other two opposite sites. | ||
| − | *ef=ac+bd | + | *ef=ac+bd (e and f are the diagonals) |
== Applicable Theorems/Formulae == | == Applicable Theorems/Formulae == | ||
Revision as of 19:05, 14 January 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, 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.
- ef=ac+bd (e and f are the diagonals)
This property is both sufficient and necessary, 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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