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?
Reference: Bizoni, M. (2017). The Golden Ratio Unexpectedly. At Right Angles, Vol. 6, No. 1, March, pp. 29-31.
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.
Some Related Links
Golden Quadrilaterals (Generalizing the concept of a golden rectangle)
The Parallel-pentagon and the Golden Ratio
A diagonal property of a Rhombus constructed from a Rectangle
An Inclusive, Hierarchical Classification of Quadrilaterals
Definitions and some Properties of Quadrilaterals
Van Aubel's Theorem and some Generalizations
A Van Aubel like property of an Equidiagonal Quadrilateral
A Van Aubel like property of an Orthodiagonal Quadrilateral
Quadrilateral Balancing Theorem: Another 'Van Aubel-like' theorem
Theorem of Gusić & Mladinić
Pitot's Theorem for a tangential or circumscribed quadrilateral
Japanese Circumscribed Quadrilateral Theorem
Perpendicular-Bisectors of Tangential Quadrilateral
Similar Parallelograms: A Generalization of a Golden Rectangle property
Extangential Quadrilateral
Area Parallelogram Partition Theorem
Area Formula for Quadrilateral in terms of its Diagonals
Crossed Quadrilateral Properties
A generalization of a Parallelogram Theorem to Parallelo-hexagons
Angle Divider Theorem for a Cyclic Quadrilateral
Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals
Bradley's Theorem for a Circumscribed Quadrilateral
Constructing a general Bicentric Quadrilateral
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Created by Michael de Villiers, 2 August 2017 with WebSketchpad; updated 12 March 2025; 2 April 2025.