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In geometry, a quadrilateral is a polygon with four sides or edges and four vertices or corners. Sometimes, the term quadrangle is used, for etymological symmetry with triangle, and sometimes tetragon for consistency with pentagon (5 sided), hexagon (6 sided) and so on. However, the term quadrangle could be considered incorrect, as the laws of Euclid state that an angle's degree measure must be less than 180, and so a concave quadrilateral has only 3 angles. The word quadrilateral can mean: In geometry, a quadrilateral is a polygon with 4 sides. ... For other uses, see Geometry (disambiguation). ... Look up polygon in Wiktionary, the free dictionary. ... A triangle. ... Look up pentagon in Wiktionary, the free dictionary. ... For other uses, see Hexagon (disambiguation). ... For other uses, see Euclid (disambiguation). ... In geometry, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions. ...

Quadrilaterals are either simple (not self-intersecting) or complex (self-intersecting). Simple quadrilaterals are either convex or concave. A simple concave hexagon In geometry, two edges of a polygon may cross or even overlap in general. ... Categories: Stub | Polygons ... A convex pentagon In geometry, a convex polygon is a simple polygon whose interior is a convex set. ... In geometry, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions. ...

Contents

Convex quadrilaterals are further classified as follows:

• Trapezium (British English) or trapezoid (Amer.): two opposite sides are parallel.
• Isosceles trapezium (Brit.) or isosceles trapezoid (Amer.): two opposite sides are parallel, the two other sides are of equal length, and the two ends of each parallel side have equal angles. This implies that the diagonals are of equal length.
• Trapezium (Amer.): no sides are parallel.
• Parallelogram: both pairs of opposite sides are parallel. This implies that opposite sides are of equal length, opposite angles are equal, and the diagonals bisect each other.
• Kite: two adjacent sides are of equal length and the other two sides also of equal length. This implies that one set of opposite angles is equal, and that one diagonal perpendicularly bisects the other. (It is common, especially in the discussions on plane tessellations, to refer to a concave kite as a dart or arrowhead.)
• Rhombus or rhomb: all four sides are of equal length. This implies that opposite sides are parallel, opposite angles are equal, and the diagonals perpendicularly bisect each other.
• Rhomboid: a parallelogram in which adjacent sides are of unequal lengths and angles are oblique (not right angles).
• Rectangle (or Oblong): all four angles are right angles. This implies that opposite sides are parallel and of equal length, and the diagonals bisect each other and are equal in length.
• Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are equal (equiangular), with each angle a right angle. This implies that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle.
• Cyclic quadrilateral: the four vertices lie on a circumscribed circle.
• Tangential quadrilateral: the four edges are tangential to an inscribed circle. Another term for a tangential polygon is inscriptible.
• Bicentric quadrilateral: both cyclic and tangential.

• An arrowhead has bilateral symmetry like a kite, but the top concaves inwards.
• An equiangular quadrilateral is a rectangle if convex, and an "angular eight" with corners on a rectangle if non-convex.
• A quadrilateral whose vertices do not all lie in a flat plane is a skew quadrilateral.

In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. ...

Taxonomy

A taxonomy of quadrilaterals is illustrated by the following graph. Lower forms are special cases of higher forms. Note that "trapezium" here is referring to the British definition (the American equivalent is a trapezoid). For the science of classifying living things, see alpha taxonomy. ...

Results from FactBites:

 The Complete Quadrilateral (1612 words) The mathematical object that goes by the name of complete quadrilateral is neither complete nor quadrilateral, at least not in the sense in which the word "quadrilateral" appears in, say, Brahmagupta's theorem about cyclic quadrilaterals. A complete quadrilateral is a configuration of four straight lines in general position and six points at which the lines intersect. First of all, we have the Theorem of Complete Quadrilateral: the midpoints of the three diagonals are collinear.
 PlanetMath: cyclic quadrilateral (170 words) A quadrilateral is cyclic when its four vertices lie on a circle. A necessary and sufficient condition for a quadrilateral to be cyclic, is that the sum of a pair of opposite angles be equal to This is version 4 of cyclic quadrilateral, born on 2001-10-06, modified 2002-03-09.
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