New Golden Ratio Construct: which one of my constructs is superior/simplest--squares & circles or just circles?

Question

I have found yet another golden ratio construction. Geogebra gives it the value of 1.61803398874990 to the ratio between the yellow and blue lines in the figure below, which is the golden ratio PHI. :) Have you seen these constructions anywhere else? And a geometric/trigonometric proof would be great for any/all of these! :) Which is the best/your favorite?

Which way do you think is the simplest/best way to construct the golden ratio in this case?

The first construction: 1. Draw a square. 2. Draw a circle with diameter the side of a square. 3. Draw a circle of the same diameter on a far corner of the square. 4. Draw a line through the centers of the circles, and the golden cut appears as drawn.

Second construction: 1. Draw a circle. 2. Draw a circle of equal diameter tangent to it on the right. 3. Draw a third circle on a line perpendicular to the line connecting the centers of the first two circles so that the third circle's center intersects with the circumference of the first circle. 4. Draw a line through the centers of the first and third circles, and the golden cut appears as drawn.

Third construction: 1. Draw a circle. 2. Draw a second circle with the same diameter centered on the circumference of the first circle. 3. Draw a third circle directly below the second circle with the same diameter centered on the circumference of the second circle. 4. Draw a fourth circle directly to the right of the third circle with the same diameter centered on the circumference of the third circle. 5. Draw a line through the centers of the first and fourth circles, and the golden cut appears as drawn.

I like this construction (these constructions) because a) the golden cut is defined by the actual shapes drawn, instead of a line added in, and b) the final step is just drawing a line through the centers of the first and last circle drawn. Pretty simple!

Have you seen these constructions anywhere else? And a geometric/trigonometric proof would be great for any/all of these! :) Which is the best/your favorite?

And perhaps you may even see a simpler way to construct it!

Answer 1

By the "secant-tangent" aspect of the Power of a Point theorem, we have $$|\overline{AB}||\overline{AC}| = |\overline{AD}|^2 \quad\to\quad q\cdot(p+q) = p^2 \quad\to\quad \frac{p}{q} = \frac{p+q}{p} = \frac{|\overline{BC}|}{|\overline{AB}|} $$

The relation between $p$ and $q$ says exactly that $\frac{p}{q}$ is the golden ratio, $1.618\dots$.

(Compare this answer.)

Answer 2

The first construction can be shown to given the golden ratio by considering ($a$ is the length of the sides of the square):

$$\phi=\frac{a}{\sqrt{(a/2)^2+a^2}-a/2}=\frac{1+\sqrt{5}}{2}$$

The second construction is exactly the same as the first. I will leave it to you to think about. As for the third, it looks to be the same as the first two as well, but the picture is a bit convoluted. I would say the first is superior because it makes clear what you are doing with the square! However, this is only an opinion.