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 rhombus' in different ways as shown below.
Definition 1: A golden rhombus is a rhombus with its diagonals in the golden ratio φ.
Apart from directly constructing a golden rhombus by starting with the construction of perpendicular, bisecting diagonals in the golden ratio φ, a golden rhombus having these properties can also easily be constructed from a golden rectangle as follows:
Construction 1a: Construct tangents to the circumcircle of a given golden rectangle EFGH at its vertices as shown below.
Challenge 1a: Prove that the diagonals AC and BD of the formed rhombus ABCD in the construction above are in the golden ratio φ (as experimentally shown by the measurements in the sketch below).
Further Challenge 1a: Prove that the tan of the angle formed by the diagonals of the golden rectangle as shown in the diagram equals 2; i.e. tan(∠EKF) = 2.
Some Golden Rhombus Constructions
Construction 1b: Given a golden rectangle EFGH, then the rhombus ABCD formed by the midpoints of its sides is a golden rhombus. Click on the Link to Construction 1b button to view.
Challenge 1b (Very easy): Prove that ABCD is a rhombus with diagonals in the golden ratio φ.
Definition 2: A golden rhombus is composed of a 'golden triangle' - which is an isosceles triangle with angles 36°, 72°, 72° - reflected in its shorter side. Click on the Link to Construction 2 button to view.
Challenge 2: Prove that the sides of the golden triangle are in the golden ratio; i.e. that BC/AC = CD/AC = φ (as experimentally shown by the measurements in the sketch above).
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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, 16 February 2022 with WebSketchpad.