Classification of Quadrilaterals

Quadrilateral is a geometric shape that consists of four points (vertices) sequentially joined by straight line segments (sides). We find the etymology of the word in S. Schwartzman's The Words of Mathematics:

The seldom used term quadrangle has exactly the same meaning as quadrilateral, however the two related terms -- complete quadrangle and complete quadrilateral -- describe essentially different configurations.

A quadrilateral may be convex or concave (see the diagram below.) A quadrilateral that is concave has an angle exceeding 180^o. In either case, the quadrilateral is simple, which means that the four sides of the quadrilateral only meet at the vertices, two at a time. So that two non-adjacent sides do not cross. A quadrilateral that is not simple is also known as self-intersecting to indicate that a pair of his non-adjacent sides intersect.

The point of intersection of the sides is not considered a vertex of the quadrilateral.

The shapes of elementary geometry are invariably convex. Starting with the most regular quadrilateral, namely, the square, we shall define other shapes by relaxing its properties.

In a square, rectangle, or rhombus, the opposite side lines are parallel. A quadrilateral with the opposite side lines parallel is known as a parallelogram. If only one pair of opposite sides is required to be parallel, the shape is a trapezoid. A trapezoid, in which the non-parallel sides are equal in length, is called isosceles. A quadrilateral with two separate pairs of equal adjacent sides is commonly called a kite. However, if the kite is concave, a dart is a more appropriate term. Kite and dart are examples of orthodiagonal quadrilaterals, i.e. quadrilaterals with perpendicular diagonals. A square and a rhombus are also particular cases of this class.

The four vertices of a quadrilateral may be concyclic, i.e., lie on the same circle. In this case, the quadrilateral is known as circumscritptible or, simpler, cyclic. If a quadrilateral admits an incircle that touches all four of its sides (or more generally, side lines), it is known as inscriptible. A quadrilateral, both cyclic and inscriptible, is bicentric.

(In the applet below, you can drag the vertices and the sides of the quadrilateral. You can display its diagonals, angle bisectors and the perpendicular bisectors of its sides. With these props, it's a simple matter to observe every single kind of quadrilateral, with a possible exception of bicentric. Which, too, is not overly difficult if you first get an isosceles trapezoid.)

As in the classification of triangles, the definitions may be either inclusive or exclusive. For example, trapezoid may be defined inclusively as a quadrilateral with a pair of parallel opposite sides, or exclusively as a quadrilateral with exactly one such pair. In the former case, parallelogram is a trapezoid, in the latter, it is not. Similarly, a square may or may not be a rectangle or a rhombus. My preference is with the inclusive approach. For, I'd like to think of a square as a rhombus with right angles, or as a rectangle with all four sides equal.

Here is a list of all the properties of quadrilaterals that we have mentioned along with the classes of the quadrilaterals that possess those properties:

Orthodiagonal or inscriptible parallelogram is a rhombus; cyclic parallelogram is a rectangle. And not to forget, every simple quadrilateral tiles the plane.

Nathan Bowler suggested a general construction of a quadrilateral with a pair of equal opposite sides and a pair of equal opposite angles which is not necessarily a parallelogram (there is a dynamic illustration):

For an isosceles trapezoid ABCD with AB = CD, the quadrilateral ABDC has a pair of equal opposite sides and two pairs of equal opposite angles.

References

1. S. Schwartzman, The Words of Mathematics, MAA, 1994