An Inclusive, Hierarchical Classification of Quadrilaterals

The quadrilaterals in the classification below display a side-angle symmetry, which is discussed more fully in my book Some Adventures in Euclidean Geometry, and is now available to download as a FREE PDF (click preceding link). Excerpts from my book showing the classification below for convex, concave as well as crossed quadrilaterals, together with their definitions, is available to download as a PDF at Inclusive, Hierarchical Classification of Quadrilaterals Excerpt.

All quadrilaterals are dynamic and can be dragged into the special cases below them, which INHERIT ALL the properties of the linked general cases ABOVE them. However, depending on how the quadrilaterals were constructed in this particular sketch there may be some limitations regarding the different shapes they can be dragged into. Click on Definitions of the Quadrilaterals, and some properties to open a separate, new window with a list of possible definitions and some important properties of these quadrilaterals.

Inclusive, Hierarchical Classification of Quadrilaterals

The classification is too large to fit onto one WebSketchpad webpage and has been divided into two webpages. Click on the LINK button in the sketch to navigate either lower down or higher up in the hierarchy.

Important Correction: I'm indebted to Martin Josefsson (2016) for pointing out that I mistakenly in the above classification assigned the 'bisecting quadrilateral' (or bisect-diagonal quadrilateral) as the side-angle dual to the trapezium (trapezoid). The correct side-angle dual to the trapezium is the extangential quadrilateral (a quadrilateral with an excircle). The side-angle duality is clearly apparent in that a trapezium has two (distinct) sums of adjacent angles equal (e.g. ∠A + ∠B = ∠C + ∠D) while an extangential quadrilateral has two (distinct) sums of adjacent sides equal (e.g. a + b = c + d). Read his excellent paper at On the classification of convex quadrilaterals.
However, in terms of an observed diagonal-bimedian duality/symmetry among quadrilaterals, the 'bisecting quadrilateral', or 'bisect-diagonal quadrilateral' as named by Joseffson (2017), can be placed as the dual to the trapezium. Read his interesting paper in this regard at Properties of bisect-diagonal quadrilaterals. The properties of a bisect-diagonal quadrilateral are further discussed in my 2021 paper Some more properties of the bisect-diagonal quadrilateral.

Notes:
a) The above classification is by no means intended as an exhaustive, complete classification of all possible quadrilaterals. One can easily add others such as the self-dual ortho-equidiagonal quadrilateral; in words, a quadilateral with perpendicular and equal diagonals. Similarly, the isosceles trapezoid (trapezium) with perpendicular diagonals is dual to the kite with equal diagonals, and both are special cases of an ortho-equidiagonal quadrilateral. An interesting, easy to prove property of these quadrilaterals are that the midpoints of their sides form a square.
b) Another example is that of the right kite and isosceles circum trapezoid (trapezium), which are both so-called bicentric quadrilaterals; in other words, subsets of those general, self-dual quadrilaterals which are both cyclic and circumscribed (but not shown in the classification scheme above). Bicentric quadrilaterals also have several interesting properties which can be found in books on advanced Euclidean geometry and mathematical journals. See Bicentric Quadrilateral Properties for some properties.
c) A zipped Sketchpad 5 sketch of this Hierarchical Classification Tree (with the exception of the Triangular Kite, Right Kite, Isosceles Circum Trapezoid and Trilateral Trapezoid) has Property buttons in the sketch, and can be downloaded from Hierarchical Quadrilateral Tree Sketch.
d) A suggested series of learning activities to visually introduce young children to the quadrilaterals and foster the idea of hierarchical class inclusion from the beginning is available at Classifying Quadrilaterals Visually.

1) Though definitions are to some extent arbitrary and we can often choose them as we wish in mathematics, it's customary to choose them for convenience. In addition, it's advisable to NOT just provide 'ready-made' definitions to students, but to involve students in actually defining and classifying the quadrilaterals themselves (e.g. see my PME paper De Villiers, 1998).

2) Sound pedagogical practice would also involve a classroom discussion and critical comparison of the merits and demerits of different possible definitions for the quadrilaterals, including carefully comparing hierarchical (inclusive) and partition (exclusive) definitions (e.g. see De Villiers, 1994; De Villiers, 2009, pp. 13-28. and Jim King, undated.) A similar analysis and approach is advocated in the 2007 book The Classification of Quadrilaterals: A Study of Definition. by Zalman Usiskin and Jennifer Griffin, and is also highly recommended.

3) Matthew Tydd from Australia has also argued in the AMESA KZN Mathematics Journal for a classification similar to the above, based on diagonal properties and symmetry. Read his paper at: Two Kites (2005).

4) Also read my 2011 paper Simply Symmetric, which briefly discusses and illustrates the value of symmetry in the choice of definitions for quadrilaterals (and some of its advantages over traditional definitions).

5) Karakonstantis & Patronis (2010) have advocated the use of a Boolean lattice for classifying quadrilaterals in their paper Relational understanding and paths of reasoning through a Boolean lattice classification of quadrilaterals.

6) Martin Josefsson has extensively explored the duality between orthodiagonal and equidiagonal quadrilaterals, as well as their characteristics, in papers published in Forum Geometricorum. Download them respectively at Characterizations of Orthodiagonal Quadrilaterals (2012) and Properties of Equidiagonal Quadrilaterals (2014).

7) In 2021, Mario Dalcín explicitly used the 'side-angle' duality in producing a direct proof of the connverse of a well-known cyclic quadrilateral theorem in his published note The side-angle duality in geometry: a direct proof of sufficiency of a cyclic quadrilateral theorem in the Mathematical Gazette.

8) Mario Dalcín in 2022, did a comprehensive investigation and systematic classification of quadrilaterals (which he more generally calls 'tetragons'), incorporating various dualities, in his paper A New Classification of Convex Tetragons.

9) In 2023, Hans Humenberger in his paper Unusual Cyclic and Tangential Quadrilaterals – An Overview in Mathematics in School explored the different types of quadrilaterals that have incircles or excircles, and identified 6 different types, including concave and crossed cases.

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Michael de Villiers, created 2011. Most recent updates: 10 August 2020; 7 September 2022; 12 March 2025; 2 April 2025.