Why force the spiral circle approach? since each explains the other, the al­ter­na­tion is progressive; it is a cyclic expression based on the symmetrical flat circle, although this represents it in a more static way.
While the square then rises the function of measuring unit and module also in e­volv­ing spiral of the golden section, virtually re­flect­ing in its same quarter for the area but also for the implications of the side (¼ of the perimeter and radius of the arcs), and thus attracting to it the quarter of a cir­cle instead of the whole,
[our] triangle seems instead to perform a function of catalyst between the circle and the square, being itself totally and directly, without additions or der­i­va­tions, 100% Golden Section, whereas the square is only its indirect container.
In fact, the two sides of the triangle matching two quarters of a circumference, are like the perfect expansion of its base at φ: it is the unit that splits; or vice ver­sa half of the base is Φ of each side.
The most significant finding is that only through this triangle, having developed the golden section thanks to the square, is it possible to trace the properties of the circle. AND IT IS THE TRI­AN­GLE THAT MAKES IT POS­SI­BLE EVEN THE SQUAR­ING.

Awaiting trial

I know well that all this may not be shared by many, or all together; but it is now undeniable that the Divine Proportion is at the root of tangible reality.
It is undoubtedly the Golden Ratio that initiates and maintains the essential work, all the more so as regards the circle – if we appreciate the evidence of how the panel of the arcs outlined from the beginning relates to it.
There is no π endowed with phantom tran­scend­ence – unless by tran­scend­ence we mean the dead end to which its current definition has landed with an endless suc­ces­sion of almost completely useless figures.
What are they good for if they can't be integrated into calculations?
It is fundamental to take note that the problem of tran­scend­ence denounced by Lindemann, and which everyone boasts as if it were a way to reinforce the def­i­ni­tion of the π, actually not only precludes the manual squaring of the circle, but equally applies to its numerical function, precisely because it cannot be re­presented in any algebraic equation with rational coefficients.
Put simply, while we will always be able to insert the value of Φ as (5±1)/2, giving the calculation interactions of future processors an increasingly precise result, in the case of π we will have to content ourselves with inserting in a ru­di­men­ta­ry way a pre-established constant, which for the most part is com­fort­a­bly reduced to 3.14 and whose digits beyond the third remain to be discussed; so much so that there is no way to verify their actual correspondence; and isn't this perhaps the greatest operational paradox? It is strange that Lindemann did not intend to make this explicit.

Incidentally, if it's any consolation, not even Leonardo da Vinci was able to fully apply the golden ratio to his Man of Vitruvius (a false ideographic examined and re­solved in 2019, in the second part of the special site dedicated to the Great Tri­an­gle); and even less the relationship between the resulting circle and square, both geometric and symbolic – although it seems to be honored by that as if it were a hologram projected into space – in an emblematic function almost op­po­site to what it should have been.
With all due grasp to the now universal applause and so much cognitive lightness.