A diagonal property of a Rhombus constructed from a Rectangle

From any given rectangle EBFD, a rhombus ABCD can be cut off or constructed as shown below by ensuring AB = AD.

Can you figure out how to do this construction, i.e. specifically how to accurately locate points A and C?


Conjecture: What do you notice about the ratio of the sides of the rectangle in relation to the ratio of the diagonals of the rhombus? Check your observation by dragging any of the RED vertices/points.

Challenge: 1) Can you explain why (prove that) your conjecture above is true?
2) Can you perhaps prove it in a slightly different way?
3) Can you perhaps find other ways of constructing a rhombus from a rectangle, or vice versa, a rectangle from a rhombus to show this side-diagonal property? Check your conjectures by clicking on the Link buttons on the bottom right to view some other possibilities as discussed in De Villiers (2017).

Note: When the rectangle in each of the three constructions becomes a 'golden rectangle', in other words, its sides are in the so-called 'golden ratio', (1 + √5)/2, then the corresponding rhombus is called a 'golden rhombus'. From the above investigation, what defining property does a golden rhombus have?

Reference: De Villiers, M. (2017). An Example of Constructive Defining: From a Golden Rectangle to Golden Quadrilaterals and beyond. At Right Angles, 6(1), March, pp. 64-69.

Read the short joint paper with my good friend & colleague Jim Metz from Hawaii A Diagonal Property of a Rhombus Constructed from a Rectangle in Learning & Teaching Mathematics, Dec 2018.


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Michael de Villiers, created 9 August 2018; updated Dec 2018.