The concept of a 'golden rectangle' - a rectangle with its sides in the 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 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). 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 Constructions
Definition 2: A golden isosceles trapezium 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 my 2019 paper in the Learning & Teaching Mathematics journal at Tiling with a Trilateral Trapezium and Penrose Tiles.
Definition 3: A golden isosceles trapezium 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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Other Golden Quadrilaterals: To view & interact with other golden quadrilaterals click here.
Paper: Read my 2017 paper An Example of Constructive Defining: From a Golden Rectangle to Golden Quadrilaterals in the journal At Right Angles.
Created by Michael de Villiers, 22 February 2022 with WebSketchpad.