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Other uses: Quadrilateral (disambiguation)
In geometry, a quadrilateral is a polygon with four sides and four vertices. Occasionally, the term quadrangle is also used, for etymological symmetry with triangle, pentagon, etc.
Classification
Quadrilaterals are either simple (not self-intersecting) or complex (self-intersecting). Simple quadrilaterals are either convex or concave. Convex quadrilaterals are further classified as follows:
- Trapezium (American English): no sides are parallel.
- Trapezium (British English) or trapezoid (Amer.): one pair of opposite sides is parallel.
- Isosceles trapezium (Brit.) or isosceles trapezoid (Amer.): two of the opposite sides are parallel, the two other sides are equal, and the two ends of each parallel side have equal angles. This implies that the diagonals have equal length.
- Parallelogram: both pairs of opposite sides are parallel. This implies that opposite sides have equal length, opposite angles are equal, and the diagonals bisect each other.
- Kite: two adjacent sides have equal length, the other two sides have equal length. This implies that one set of opposite angles is equal, and that one diagonal perpendicularly bisects the other.
- Rhombus: four sides have equal length. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other.
- Rectangle: each angle is a right angle. This implies that opposite sides are parallel and have equal length, and the diagonals bisect each other and have equal length.
- Square (regular quadrilateral): four sides have equal length, and each angle is a right angle. This implies that opposite sides are parallel, and that the diagonals perpendicularly bisect each other and are of equal length.
- Cyclic quadrilateral: the four vertices lie on a circumscribed circle.
- Tangential quadrilateral: the four edges are tangential to an inscribed circle.
- Bicentric quadrilateral: both cyclic and tangential.
Is a square a rectangle?
Some people define categories exclusively, so that a rectangle is a quadrilateral with four right angles that is not a square. This is appropriate for everyday use of the words, as people typically use the less specific word only when the more specific word will not do.
But in mathematics, it is important to define categories inclusively, so that a square is a rectangle. Inclusive categories make statements of theorems shorter, by eliminating the need for tedious listing of cases. For example, the visual proof that vector addition is commutative is known as the "parallelogram diagram". If categories were exclusive it would have to be known as the "parallelogram (or rectangle or rhombus or square) diagram"!
Compare the question of whether a real number is a complex number.
Taxonomic classification
The taxonomic classification of quadrilaterals is illustrated by the following graph. Lower forms are special cases of higher forms.
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