Cyclic polygon: Difference between revisions
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Every [[triangle]] is cyclic, since any three (non-[[collinearity|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> | |||
Revision as of 13:26, 25 November 2008
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:
