"... the algorithmically constructive and creative definition ... models new objects out of familiar ones." - Hans Freudenthal (1973:458) in Mathematics as an Educational Task.
The concept of a 'golden rectangle' - a rectangle with its sides in the golden ratio φ = (√5 + 1)/2 - can be extended to that of a 'golden isosceles trapezium' in different ways as shown below.
Definition 1: A golden isosceles trapezium (trapezoid) can be constructed from a 'golden parallelogram - Definition 1' - by the 'addition' or the 'subtraction' of an equilateral triangle on one of the shorter sides of the golden parallelogram.
a) The dynamic sketch below shows the 'addition' of an equilateral triangle to a golden parallelogram (which has its sides and diagonals in the ratio φ).
b) Click on the Link to Construction 2' button to view to view the 'subtraction' case.
Challenge 1
a) Prove for both cases (constructions 1 & 2) that the sides of the golden isosceles trapezium ABCD, defined above & illustrated below, form a geometric progression in terms of φ from the shortest to the longest side; i.e. in the 1st case, prove that AD/AB = BC/AD = φ; in the 2nd case, prove that AB/BC = AD/AB = φ.
b) Prove that in the 1st case, the diagonals are divided in the ratio φ; i.e. CE/EA = φ.
c) Prove that in the 2nd case, the diagonals are divided in the ratio φ2; i.e. AE/EC = φ2.
Some Golden Isosceles Trapezium (Trapezoid) Constructions
Definition 2: A golden isosceles trapezium (trapezoid) can be obtained by reflecting a 'golden triangle' - which is an isosceles triangle with angles 36°, 72°, 72° - around the perpendicular bisector of one of its longer sides. Click on the Link to Construction 3' button to view.
Challenge 2
a) Prove that the sides of the golden triangle above are in the golden ratio; i.e. that BC/AB = φ (as experimentally shown by the measurements in the sketch).
b) Show that BC/AB = CE/EA = φ.
Note that this golden isosceles trapezium is a special 'trilateral trapezium', i.e. having 3 equal sides, and that the 'base' angles are bisected by the diagonals.
c) Can you create a tiling with this golden isosceles trapezium and a penrose kite or dart? Investigate before checking & comparing your findings with those in De Villiers (2018).
Definition 3: A golden isosceles trapezium (trapezoid) can be obtained by giving a 'golden triangle' - which is an isosceles triangle with angles 36°, 72°, 72° - a halfturn around the midpoint of one of its longer sides; and then repeating the same halfturn on the image. Click on the Link to Construction 4' button to view.
a) Obviously from the construction AB/AD = φ. Also note that compared to the preceding case, this golden isosceles trapezium appears a little 'thinner' & 'taller'.
Challenge 3
With reference to the sketch, show that the diagonals divide each other in the ratio 2 to 1; i.e. CE/EA = 2.
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Critical Comparison
Critically compare the possible definitions of a 'golden isosceles trapezium' above, judging them not only by 'appearance' or 'shape', but also by their mathematical properties & the ease by which one can construct an example or deduce other additional properties (not contained in the definition).
a) Which one of these do you most prefer? Why?
Other Golden Quadrilaterals
To view, interact with & explore other golden quadrilaterals CLICK HERE.
Constructive Defining
Read more about constructive (and descriptive) defining and classifying in mathematics in De Villiers (2009).
References
De Villiers, M. (1994, 1996, 2009). Defining and Classifying (Chapter 2), pp. 13-28, from Some Adventures in Euclidean Geometry. (Free to download). Morrisville, NC: Lulu Publishers, Dynamic Mathematics Learning.
De Villiers, M. (2017). An Example of Constructive Defining: From a Golden Rectangle to Golden Quadrilaterals. At Right Angles, Vol. 6, No. 1, pp. 64-69; Vol. 6, No. 2, pp. 74-81.
De Villiers, M. (2018). Tiling with a Trilateral Trapezium and Penrose Tiles. Learning & Teaching Mathematics, No. 25, pp. 6-10.
Related Links
Golden Parallelogram (involves golden ratio φ)
Golden Quadrilaterals (generalizing a 'golden rectangle')
Exploring Isosceles Trapezoid Properties (Rethinking Proof activity)
Isosceles Trapezoid Midpoints (Rethinking Proof activity)
Free Construction of an Isosceles Trapezoid
Systematizing Isosceles Trapezoid Properties - Defining an Isosceles Trapezoid (Rethinking Proof activity)
Some Properties of Bicentric Isosceles Trapezia & Kites
Alternate sides cyclic-2n-gons and Alternate angles circum-2n-gons
(Generalizations of isosceles trapezia and kites)
Some Trapezoid (Trapezium) Explorations
Midpoint trapezium (trapezoid) theorem generalized
Investigation 4: Defining Quadrilaterals (Grades 8-10)
An Inclusive, Hierarchical Classification of Quadrilaterals
Some Possible Definitions of Quadrilaterals & some Properties (Relates to the above classification)
Similar Parallelograms: A Generalization of a Golden Rectangle property
The Parallel-pentagon and the Golden Ratio
Cyclic Kepler Quadrilateral and the Golden Ratio
A diagonal property of a Rhombus constructed from a Rectangle
Investigating Centres of Isosceles Trapezia & Kites coinciding with their Centroids
External Links
Golden rectangle (Wikipedia)
Golden ratio (Wikipedia)
Golden triangle (Wikipedia)
Isosceles trapezoid (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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Created by Michael de Villiers, 22 February 2022 with WebSketchpad; updated 19 April 2026.