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?

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.