The below Golden Ratio Construction results in a ratio of PHI (1.6180...) between the blue line and red line, as found in Geogebra. This seems like a simple construction of the golden ratio, yet so far I have not found anything similar. If you know of a similar golden ratio construction, please do share! Thanks!
This Golden Ratio construction is made in the following manner:
- Draw two adjacent squares Square1 and Square2.
- Draw a circle with radii equal to the side of the Square, placed at the corner of Square2 as drawn below.
- A line passing through the two opposing corners of the two adjacent squares will then define the golden cut in conjunction with the circle, as shown below. The ratio of segment t to segment s, (or the blue segment to the red segment), is the golden ratio PHI.
Has anyone come across anything the same or similar?
And of course trigonometric and geometric proofs of the golden ratio construction are always welcome!
Your construction in this question occurs as a subobject in the construction here. Your particular result does not appear explicitly there, but is produced by projecting Molokach's points $A$, $B$, and $H$ through an unconstructed point interior to the triangle $ACH$.
Your duplicate question, New, extremely simple golden ratio construction with two identical circles and line. Is there any prior art?, using the two circles construction is equivalent to the left pair of the three circles construction here.