In the latter case we can switch to a direct execution of the much coveted quad­ra­ture, re-starting from the basic square, (side Φ of the radius Ca in the figure) bypassing the preliminary construction of the great triangle with its whole co­hort, which in any case remains the keystone in translating the tran­scend­ence of the π into immanent form.

According to the procedure de­scribed above, from the base of the starting square with vertex in C we will obtain the length Ca in in­creas­ing golden proportion φ, of which we will make the radius of the primary circle, assigning it a val­ue of 1,0 (instead of the di­am­e­ter, whereby the Φ0 in the fig­ure).
Therefore the side of the square C will take on value Φ.
To obtain its square root it will be sufficient to extend it from C to b, and then with the center in the middle of ba to trace the semicircle apb [green stroke] which will be intersected at the point p by the primary diameter, as a ver­ti­cal extension passing through C.
The Pythagorean relation is confirmed bC:Cp = Cp:1, from which Cp = bC, side of ¼ of the square whose perimeter has the length of the circumference with radius Ca.

Similarly we proceed from this by adding it in turn on the diameter to the ra­di­us Ca thanks to the arc pd with center in C, then tracing the semicircle dqa [or­ange stroke] with a center in the middle of da, intersected by the pri­ma­ry ver­ti­cal di­am­e­ter at point q, so that Cq, the square root of Cd and Cp, will be the side of the ¼ of the square with the same area as the circumference with radius Ca = 1.
It will be enough to reproduce the last square obtained, superimposing it on the circle in the four quadrants to take note of its compliance without compromise.

If we then wanted to quarter the vision on a functional value of π as sim­ple2Φ, we would likewise see in it the side of the square with a perimeter e­quiv­a­lent to the circumference with radius = ½, and in the form4Φ the side of the square with an area matching that of the circle with radius = ½: a kinship at the DNA level.
I don't think you can ask for more for squaring the circle.

Summing up

The writer will not take a position between the three perspectives, nor seek glory for himself, he has already collected enough of it in the course of previous incarnations. Nor does it expect profits: its seven Web domains, containing several searches and advanced solutions (see below), do not sell ser­vic­es or ad­ver­tise­ments of any kind.
As long as it is not said that "he came close"...
Initially he would have been satisfied if everyone had intended to agree that this sacred triangle was a subject of such importance as not be kept closed in a drawer, whatever the outcome and the risk might be; but now that his research has matured, he no longer has any doubts.