In my 1990 paper on proof the 'discovery' function of proof was illustrated with an example of where further reflection on an 'explanatory' proof leads to a new more general result. This illustrates that understanding why a result is true is an added benefit of proof. Below is another example of an elementary result that can easily be used in the classroom to illustrate this discovery function of proof.
Area Parallelogram Partition Theorem
The following lovely little result is given in a paper by Philippe Richard (2003) and was also used as a problem in one of the Intermediate Olympiads in the UK (Jobbings, 2013):
Given a parallelogram ABCD and arbitrary points E and F, then as shown below, Area BGE + Area IFD + Area ECFH = Area AGHI.
Select and drag any of A, B, C, E or F to dynamically change the figure and investigate the result.
References
De Villiers, M. (1990). "The Role and Function of Proof", Pythagoras, 24, 17-24.
Jobbings, A. (2013). A Problem-Solver's Handbook: A Guide to Intermediate Mathematical Olympiads. The United Kingdom Mathematics Trust, University of Leeds, UK.
Richard, P.R. (2003). "Proof Without Words: Equal Areas in a Partition of a Parallelogram", Mathematics Magazine, Dec. 2003, p. 348.
Area Parallelogram Partition Theorem
Challenge
1) Can you explain why (prove that) the above result is true?
2) Click on the Show Hint button if necessary.
3) If still stuck, first try this simpler, auxiliary problem "Simpler Partition of a Parallelogram", to prove the equality of areas EGH + DIC = AGD + HIF.
4) Carefully reflect on your proof(s), and consider how this same proof can also apply to a certain type of pentagon, hexagon, etc. Make generalizations and check your generalized conjectures by clicking on the Link buttons on the right to go to pentagons, hexagons, etc. with a similar area partition property.
5) Can you explain (prove) these generalizations as well?
6) Is it necessary for the area partition result to hold for a 2n-gon with n > 2 that the 2n-gon must have all pairs of opposite sides equal? Explain why, or why not?
Published Paper
Read my short 2019 (online) & 2020 (hard copy) paper "Proof as a means of Discovery: Area parallelogram partition theorem" in IJMEST.
Related Links
An Area Preserving Transformation: Shearing
A Surprising Constant Area Sum Involving Translating Figures
Area Ratios (Rethinking Proof activity)
Varignon Area (Rethinking Proof activity)
International Mathematical Talent Search (IMTS) Problem Generalized
A Geometric Paradox Explained (Another variation of an IMTS problem)
Another parallelogram area ratio
Parallelogram Distances & Areas (Click 'Link to Areas')
Vreken's sum of triangular areas in a regular hexagon
Area ratios of some polygons inscribed in quadrilaterals and triangles
Feynman Parallelogram Generalization
Feynman's Triangle: Some Generalizations & Variations
Area Formula for Quadrilateral in terms of its Diagonals
Sylvie's Quadrilateral Area Ratio Theorem
Finding the Area of a Crossed Quadrilateral (PDF)
Crossed Quadrilateral Properties
Bretschneider's Quadrilateral Area Formula & Brahmagupta's Formula
The Equi-partitioning Point of a Quadrilateral
Free Download of Geometer's Sketchpad & Associated Learning/Instructional Modules on various topics
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Michael de Villiers, created with WebSketchpad, 25 July 2018; updated 10 July 2019; 2 April 2024; 30 Sept 2025; 17 April 2026; 24 August 2026.