Cyclic polygon: Difference between revisions

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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,
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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>

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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: