A recent paper by Bizony (2017) discussed some interesting golden ratio properties of a Kepler triangle, defined as a right-angle triangle with its sides in geometric progression in the ratio 1 : √φ : φ, where φ = (1 + √5)/2. This inspired me to wonder what would happen if one similarly defined a "Kepler quadrilateral" as a quadrilateral with sides in geometric progression of √φ. Would such a quadrilateral perhaps also exhibit some golden ratio, or other interesting, properties?

Below is shown such a "Kepler quadrilateral" ABCD with sides constructed in geometric progression as stated above. Note that it is 'flexible' and that it has a variable shape, e.g. drag vertices B or C. Though no immediate invariant properties of the general shape appear apparent, click on the BUTTON at the bottom to show the perpendicular bisectors of the sides, then DRAG vertices B or C until the perpendicular bisectors become concurrent, and ABCD becomes cyclic. Observe the shown measurements. What do you notice? What conjectures can you make?

Cyclic Kepler Quadrilateral Conjectures

Challenge:
Can you explain why (prove) your conjectures above are true?

Read my paper "A Cyclic Kepler Quadrilateral and the Golden Ratio" published in the March 2018 issue of the At Right Angles journal.

Note: A cyclic Kepler quadrilateral, and more more generally, any cyclic quadrilateral ABCD with its sides AB : BC : CD : DA in geometric progression with common ratio r, is a 'bisect-diagonal' quadrilateral. For more info, go to Bisect-diagonal Quadrilateral.

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Created by Michael de Villiers, 2 August 2017 with WebSketchpad.