# New Golden Ratio Construction with Two Adjacent Squares and Circle. Have you seen anything similar?

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:

1. Draw two adjacent squares Square1 and Square2.
2. Draw a circle with radii equal to the side of the Square, placed at the corner of Square2 as drawn below.
3. 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!

• I haven't seen such a construction before, but just a quick analysis - normalizing the side length of the squares to 1, we have the diagonal of the rectangle with length $\sqrt{1^2+2^2}=\sqrt{5}$, giving the red line length $\sqrt{5}-1$. The diameter of the circle (i.e. the blue line) has length $2$, so the ratio of the blue line to red line is $\frac{2}{\sqrt{5}-1}=\frac{2\left(\sqrt{5}+1\right)}{4}=\frac{\sqrt{5}+1}{2}$ as desired. Commented May 23, 2016 at 21:31
• @Peter Woolfitt That could be an answer to the follow on question.
– user301988
Commented May 24, 2016 at 0:46
• It is very similar to one of the steps for the usual construction of the regular pentagon (en.wikipedia.org/wiki/Pentagon). Commented May 24, 2016 at 2:40
• which step @JackD'Aurizio? could you please be more specific and perhaps even provide a figure or link to a figure or drawing? thanks! :) Commented May 24, 2016 at 2:41
• en.wikipedia.org/wiki/File:Richmond_pentagon_1.PNG Commented May 24, 2016 at 2:42

A similar construction is shown at http://www.goldennumber.net/circles/ in 2006. It just happens to have circles drawn on either side of the center circle, which are not important to the golden ratio point that is created.

• i count three circles used there. i only use two circles. or in this case a circle and two squares. :) Commented May 27, 2016 at 9:08

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.

• Thanks! In both cases it seems my constructs are simpler--fewer circles and less complicated objects. :) Commented May 24, 2016 at 5:43
• thanks again @EricTowers for your feedback confirming that prior art doesn't exist. have you ever seen any prior art for the new golden ratio construct i just posted here: math.stackexchange.com/questions/1798843/… ? :) Commented May 25, 2016 at 3:48