# Become Great at Mathematics for GIS Professionals

# 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*:

**quadrilateral** (noun, adjective): the first element is from Latin *quadri-* “four” from the Indo-European root *k ^{w}etwer-* “four.” The second element is from Latin

*latus*, stem

*later-*, “side,” of unknown prior origin. A quadrilateral is a four-sided polygon. The Latin term is a partial translation of Greek

*tetragon*, literally “four angles,” since a closed figure with four angles also has four sides. Although we use words like

*pentagon*and

*polygon*, the term

*quadrilateral*has completely replaced

*tetragon*.

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.

A *square* is a quadrilateral with all sides equal and all angles also equal. Angles in any quadrilateral add up to 360°. It follows that, in a square, all angles measure 90°. An *equiangular* quadrilateral, i.e. the one with all angles equal is a*rectangle*. All angles of a rectangle equal 90°. An *equilateral* quadrilateral, i.e. the one with all sides equal, is a *rhombus*.

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*.

The diagram below (which is a modification of one from wikipedia.org) summarize the relationship between various kinds of quadrilaterals:

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:

Property | Quadrilaterals | |

Orthodiagonal | Kite, Dart, Rhombus, Square | |

Cyclic | Square, Rectangle, Isosceles Trapezoid | |

Inscriptible | Kite, Dart, Rhombus, Square | |

Having two parallel sides | Rhombus, Square, Rectangle, Parallelogram, Trapezoid | |

Having two pairs of parallel sides | Rhombus, Square, Rectangle, Parallelogram | |

Equilateral | Rhombus, Square | |

Equiangular | Rectangle, Square |

Orthodiagonal or inscriptible parallelogram is a rhombus; cyclic parallelogram is a rectangle. In particular, a parallelogram with equal diagonals is necessarily a rectangle. And not to forget, every simple quadrilateral tiles the plane.

A simple quadrilateral with two pairs of equal opposite angles is a parallelogram. (Because then the opposite sides are parallel.) A simple quadrilateral with two pairs of equal opposite sides is a parallelogram. (Because of SSS when you draw one of the diagonals.)

There is a simple quadrilateral with two pairs of equal sides: a kite (or a dart). It does have a pair of opposite equal angles.

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):

Let ABC be isosceles with AB = AC. Pick D on BC. Let C’ be the reflection of C in the perpendicular bisector of AD. ABDC’ has two opposite sides the same length and two opposite angles equal but is not a parallelogram if D isn’t the midpoint of AB. This construction gives all such quadrilaterals.

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.