"... the Socratic didactician would refuse to introduce the geometrical objects by definitions, but wherever the didactic inversion prevails, deductivity starts with definitions." - Hans Freudenthal (1973:416) in Mathematics as an Educational Task.
"To me it appears a radically vicious method, certainly in geometry, if not in other subjects, to supply a child with ready-made definitions, to be subsequently memorized after being more or less carefully explained. To do this is surely to throw away deliberately one of the most valuable agents of intellectual discipline. The evolving of a workable definition by the child's own activity stimulated by appropriate questions, is both interesting and highly educational." Benchara Blandford (1908) quoted in Griffiths & Howson (1974: 216-217), Mathematics: Society and Curricula. Cambridge University Press.
The geometry learning activity below is from the "Proof as Systematization" section of my book Rethinking Proof (free to download).
Worksheet & Teacher Notes
Open (and/or download) a guided worksheet and teacher notes to use together with the dynamic sketch below at: Systematizing Rhombus Properties - Defining a Rhombus Worksheet & Teacher Notes.
Prerequisites
Knowledge of the properties of a rhombus - specifically it is expected that you have already completed the following rhombus activities:
Visually Introducing & Classifying a Rhombus (Suggested Grades 1-7)
Exploring Rhombus Properties (Suggested Grades 4 - 9)
Free Construction of a Dynamic Rhombus (Suggested Grades 6 - 10)
Short Description
In the preceding online activity Exploring Rhombus Properties, you should have discovered that a rhombus has (among others) the following important properties:
Check by Construction
2. In the dynamic sketch below you are provided with 7 different possible constructions. Press each button step by step
on each of the seven pages to construct the figures. When each construction is finished, match each page with a descriptions a - g above or in the table given in the Defining a Rhombus Worksheet.
'.....Drag each figure & use measurements in the Toolbar on the left (scroll down if necessary) to see if it always remains a rhombus. (Note: Since a rhombus can be dragged into the shape of a square, we regard a square as a special rhombus.) In the table given in the accompanying Defining a Rhombus Worksheet, cross out the names of any pages that construct quadrilaterals that are not always rhombuses.
Checking Descriptions/Definitions by Construction
3. List the descriptions from a – g that you think correctly describe (define) a rhombus.
4. State the description from a – g that you personally think best describes a rhombus. Try to defend your choice with good reasons.
Challenge
5. Start from a description (definition) of your choice as your given assumption and then prove as theorems that a rhombus has each of the other properties not listed in your description (definition). You can use any new theorems that you prove in the subsequent proofs of the other properties.
Proving Rhombus Properties from Definitions
In the preceding part of this activity, you should have found that each of the following descriptions could be used to accurately construct a rhombus:
'.....A. A rhombus is any quadrilateral with two perpendicular axes of symmetry (each through a pair of opposite angles).
'.....B. A rhombus is any quadrilateral with perpendicular, bisecting diagonals.
'.....C. A rhombus is any quadrilateral with all sides equal.
'....D. A rhombus is any quadrilateral with one pair of adjacent sides equal and opposite sides parallel.
In mathematics, we call such descriptions definitions.As we can see, there may be many different, alternative ways in which we can define mathematical objects. We now have to show that all the other properties of a rhombus
logically follow as theorems from each of these definitions. We will now give an example for Definition B.
6. Work through the guided proofs following from Definition B as given in the accompanying Defining a Rhombus Worksheet.
Explore Further Definitions
7. Now choose any two of the other three possible definitions A, C, or D for a rhombus. For each, show, as in the worked example in the Worksheet, how the remaining properties listed at the beginning and not included in your definition can be proved as theorems.
8. A concept can be defined in terms of its relationships with other concepts. A rhombus can also be viewd as a special parallelogram or a special kite, since both of these can be dragged into the shape of a rhombus (e.g. see Visually Introducing & Classifying Quadrilaterals - navigate to the parallelogram & kite). Try to define a rhombus by making use of these relationships.
9. a rhombus can also be viewd as a special circumscribed or tangential quadrilateral (that is a quadrilateral circumscribed around a circle). Try to define a rhombus as a circumsrobed (tangential) quadrilateral with additional properties.
Class Discussion
A definition can be seen as an agreement among interested parties about what a specific object is. Although you have now seen it is possible to define a rhombus in many different ways, it can be very confusing if everyone is using a different definition. It is therefore now necessary to choose a common definition that will be acceptable for the whole class.
10. Have a class discussion to decide which definition of a rhombus is most convenient for you.
Further Challenges
11. Prove the following two dual statements, namely, that
a) a parallelogram with an incircle is a rhombus
b) a parallelogram which is cyclic is a rectangle.
12. Prove that a kite with (at least) one pair of opposite sides parallel is a rhombus.
Defining and Investigating New Concepts
New mathematical objects are often defined by modifying or extending the definitions or properties of known objects in mathematics. This is known as constructive (a priori) defining.
i) For example, we can generalize the concept of a rhombus to an equilateral polygon, which is a polygon with equal sides. An example of a dynamic equilateral pentagon is given at Generalizations of Viviani's Theorem to Polygons.
ii) We can also generalize the concept of a rhombus to a semi-regular side-gon (or equilateral circumscribed 2n-gon), which can be defined as any 2n-gon with an axis of symmetry through each pair of opposite angles (vertices) - see Semi-regular Angle-gons and Side-gons.
Pedagogical Notes
a) The main purpose of this activity is to introduce students to the systematization function of proof: the fact that proof is an indispensable tool in the organization of known results into a deductive system of definitions
and theorems.
b) Further objectives are
• Developing students’ understanding of the nature of definitions as unproved assumptions, as well as the
existence of alternative definitions.
• Engaging students in the evaluation and selection of different formal, economical definitions rather than
just providing them with a single ready-made definition.
• Developing students’ ability to construct formal, economical definitions for geometrical concepts.
References
Bennett, Dan. (2012). Constructing Rhombuses, pp. 161-163. Exploring Geometry with Geometer's Sketchpad (free to download). Key Curriculum Press.
Bennett, Dan. (2012). Defining Special Quadrilaterals, pp. 143-145. Exploring Geometry with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (1998). To Teach Definitions in Geometry or Teach to Define? In A. Olivier & K. Newstead (Eds), Proceedings of the Twenty-second International Conference for the Psychology of Mathematics Education: Vol. 2. (pp. 248-255). University of Stellenbosch: Stellenbosch, 12-17 July 1998.
De Villiers, M.; Govender, R. & Patterson, N. (2009). Defining in Geometry. In T. V. Craine & R. Rubenstein (Eds.). (2009). Understanding Geometry for a Changing World: NCTM's 71st Yearbook. Reston, VA: NCTM.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
Govender, R. & De Villiers, M. (2004). A dynamic approach to quadrilateral definitions. Pythagoras, 58, June, pp. 34-45.
Other Rethinking Proof Activities
Other Rethinking Proof Activities
Related Links
Visually Introducing & Classifying a Rhombus (Suggested Grades 1-7)
Exploring Rhombus Properties (Suggested Grades 4 - 9)
Free Construction of Dynamic Rhombus (Suggested Grades 6-10)
Visually Introducing & Classifying Quadrilaterals
Exploring the Properties of (some) Quadrilaterals
Constructing (some) Dynamic Quadrilaterals
Defining (some) Quadrilaterals
Introducing, Classifying, Exploring, Constructing & Defining Quadrilaterals
Some Van Hiele theory video clips and invited papers
A Hierarchical (Inclusive) Classification of Quadrilaterals
Definitions and some Properties of Quadrilaterals
Semi-regular Angle-gons and Side-gons (Generalizations of rectangles and rhombi)
Golden Rhombus: an Example of Constructive Defining
A diagonal property of a Rhombus constructed from a Rectangle
A Rhombus Angle Trisection Result
A generalization of Paul Yiu's problem
The 120o Rhombus (or Conjoined Equilateral Triangles) Theorem
Opposite Side Quadrilateral Properties by Kalogerakis
Van Aubel's Theorem and some Generalizations
External Links
Van Hiele model (Wikipedia)
Rhombus (Wikipedia)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Michael de Villiers, created with WebSketchpad, 2 Nov 2025; updated 15 May 2026.