Cyclic polygon: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Richard Pinch
m (sp)
mNo edit summary
 
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
In [[plane geometry]], a '''cyclic polygon''' is a [[polygon]] whose vertices all lie on one [[circle]].  The centre of the circle is the [[circumcentre]] of the polygon,
{{subpages}}
In [[plane geometry]], a '''cyclic polygon''' is a [[polygon]] whose vertices all lie on one [[circle]].  The centre of the circle is the [[circumcentre]] of the polygon.


Every [[triangle]] is cyclic, since any three (non-collinear) points lie on a unique circle.
Every [[triangle]] is cyclic, since any three (non-[[collinearity|collinear]]) points lie on a unique circle.


A '''cyclic quadrilateral''' is a [[quadrilateral]] whose four vertices are concyclic.  A quadrilateral is cyclic if and only if pairs of opposite angles are [[supplementary]] (add up to 180°, π [[radian]]s).
==Cyclic qusdrilateral==
A '''cyclic quadrilateral''' is a [[quadrilateral]] whose four vertices are concyclic.  A quadrilateral is cyclic if and only if pairs of opposite angles are [[supplementary]] (add up to 180°, π [[radian]]s). '''Ptolemy's theorem''' states that in a cyclic quadrilateral ''ABCD'', the product of the diagonals is equal to the sum of the two products of the opposite sides:
 
:<math>AC \cdot BD = AB \cdot CD + BC \cdot AD .\,</math>[[Category:Suggestion Bot Tag]]

Latest revision as of 17:00, 3 August 2024

This article is developing and not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In plane geometry, a cyclic polygon is a polygon whose vertices all lie on one circle. The centre of the circle is the circumcentre of the polygon.

Every triangle is cyclic, since any three (non-collinear) points lie on a unique circle.

Cyclic qusdrilateral

A cyclic quadrilateral is a quadrilateral whose four vertices are concyclic. A quadrilateral is cyclic if and only if pairs of opposite angles are supplementary (add up to 180°, π radians). Ptolemy's theorem states that in a cyclic quadrilateral ABCD, the product of the diagonals is equal to the sum of the two products of the opposite sides: