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. | 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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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: | 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> | :<math>AC \cdot BD = AB \cdot CD + BC \cdot AD .\,</math>[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 17:00, 3 August 2024
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:
