Golden Ratio Symphony in a 5x5 Grid & Circle: Is this golden ratio construction derivative of other constructs? Is it novel?

Question

Below please enjoy a Golden Ratio Symphony in a 5x5 Grid & Circle: Is this golden ratio construction derivative of other constructs? Is it novel?

The below construction is created by beginning with a 5x5 array of squares. A circle is then inscribed passing through the corners of the smaller square defined by the center 3x3 array of squares as drawn.

Golden ratios in the above figure include:

e/f

a/b

g/h

i/j

c/d

and I reckon there are more!

This construction does not seem to be based on nor derivative of any smaller constructions. What do others think? Is it novel and unique? Have you seen it elsewhere? And per usual, geometric proofs and philosophical analysis are welcome in the discussion! :)

P.S. Here is another "Golden Ratio Symphony" construction of mine consisting of a golden ratio symphony in a 3x3 grid and squares: A Golden Ratio Symphony! Why so many golden ratios in a relatively simple golden ratio construction with square and circle?

Answer 1

Construct the tangent from the endpoint of $a$ to the circle, and taking a square's diagonal as unit length we get $a(3-a) = 1$ by power of point, and solving gives $a = 2-φ = \frac1{φ+1}$, giving $\frac{e}{f} = φ$ and $b = 1-a = \frac{φ}{φ+1} = \frac1φ$ so $\frac{b}{a} = \frac{g}{h} = φ$ and $\frac{d}{c} = \frac1b = φ$. And $\frac{i}{j} = \frac{g}{h}$ by trivial similar triangles.

Alternatively, obtain any one ratio and then obviously the others follow since $\frac{x}{y} = φ$ implies $\frac{x+y}{x} = \frac{y}{x-y} = φ$.

Frankly, you should learn how to do all this yourself instead of continually getting people from Math SE to do it for you.

Answer 2

A general pattern in how a line cuts a circle to result in $\varphi$ is given below. Reference is to several of your recent constructions. I am looking if the segments placed this way may be connected to this scheme after all..( this was posted elsewhere and deleted but now relocated here.)

If square-root of power of Circle (or tangent length T) from (external point O) equals enclosed secant length PQ inside Circle cut by transversal from O, then the segments OP, OQ bear Golden Ratios to T.

Tangent is inclined maximum at $$ \cos \theta_{together t} = (T/h) $$

The transversal has smaller inclination given buy

$$ \cos \theta_{\varphi} = \frac{\sqrt 5 T}{4 \,h} $$

Calculation

$$ h^2 = a^2 + T^2 $$

To have $ \boxed{ OQ/OT = \varphi \, ; OP/OT = 1/\varphi } ; $ with $ OP\cdot OQ = OT^2$ ( Property of Circle)

the difference of segments from O should be $ (OP-OQ) = T (\varphi -1/\varphi) = T. $

So to get the two Golden Ratios just transfer line segment T as a secant rotating it on circle boundary until it passes thorough origin.

Using Cosine Rule $ \theta_{\varphi} $ can be found from $ \Delta QOC$ resulting as above.

The constructions below have $ OT=1$ unit.

EDIT1:

A construction to obtain GoldenRatios for negative power, may be a generator for $\varphi$ is also given:

Construct a right angled triangle OTC with side OC on x-axis and OT = 1 unit on y-axis, hypotenuse CT= R as radius. Construct circle with center C and radius R. Construct a semicircle with OC as diameter. With O as center, OT/2 as radius draw a short arc to cut semicircle at G. Join OG and extend OG on either side so it cuts bigger circle centered at C at points $R_1,R_2.$

Segments $GR_1,GR_2$ are of length $\varphi_1,\varphi_2.$

Unable to give a sketch immediately (iPadding, computer kaputt), appreciate construction and uploading from anyone among you as above. Omg, it is all on hold, did not note this ...