Discovery of the true geometric π and Squaring the circle

12/₃₄	The KEY Once the four circles have been identified by their respective diameters in decreasing golden proportion, let's now move on to the first initiative, which consists in tracing from the vertical point `V` on the larger circumference, of diameter = `1`, a line tangent to the circumference Φ, which will meet the primary at point `A`. With or without algebraic proof – I produced it here in 2009 – we will immediately find out some interesting aspects: drawing the symmetrical `VB` line to `VA` and joining `A` with `B` we will have drawn a triangle whose base `AB` is tangent to the circle Φ³; therefore we would have obtained `B` also following the reverse process. a mirror line at `VA` halves the side at the point `X` which is tangency of `VA` to the circle Φ; and reaches exactly the point of tangency at Φ³ in the middle of `AB`; in particular a line parallel to `AB` from the point `X` is tangent to Φ², perhaps I should add 'obviously'; but the best is yet to come… The sum of `AV` and `VB` is equivalent to `AB×`Φ, i.e. the base of this especial triangle is the Golden Section of the sum of the sides; the supreme balancing factor in the structure of the ordered physical universe; this had since long led me to define it as the Great Golden Triangle par excellence. Driven by these premises, I forwarded to an in-depth study that proved to be decisive; and now the time has come to grapple with the numbers. Given the diameter of the primary circle as a unit of measurement, therefore with a value of `1.0`, the circumference will be `= π` and the quadrant arc `VQ = π/₄` (in a linear sense, as opposed to angular measurement in radians). In this case the `VD` height of the triangle will be `VC+CD` ie `0,5 +` Φ`³/₂` [0,2360679774997896964/2… = 0,1180339887498948482…+ 0,5], which in our construct turns out to be the golden section Φ of the diameter = 1: `0,618033988749894848204586834365638117720309179805762862135…` To measure `VA` it will be neessary according to the theorem of Pythagoras: `AD = √AC² – CD²`

12/₃₄

The KEY

Once the four circles have been identified by their respective diameters in decreasing golden proportion, let's now move on to the first initiative, which consists in tracing from the vertical point V on the larger circumference, of diameter = 1, a line tangent to the circumference Φ, which will meet the primary at point A.

With or without algebraic proof – I produced it here in 2009 – we will immediately find out some interesting aspects:

drawing the symmetrical VB line to VA and
joining A with B we will have drawn a triangle whose base AB is tangent to the circle Φ³;
therefore we would have obtained B also following the reverse process.
a mirror line at VA halves the side at the point X which is tangency of VA to the circle Φ; and reaches exactly the point of tangency at Φ³ in the middle of AB; in particular a line parallel to AB from the point X is tangent to Φ², perhaps I should add 'obviously'; but the best is yet to come…
The sum of AV and VB is equivalent to AB×Φ, i.e. the base of this especial triangle is the Golden Section of the sum of the sides; the supreme balancing factor in the structure of the ordered physical universe; this had since long led me to define it as the Great Golden Triangle par excellence.

Driven by these premises, I forwarded to an in-depth study that proved to be decisive; and now the time has come to grapple with the numbers.

Given the diameter of the primary circle as a unit of measurement, therefore with a value of 1.0, the circumference will be = π and the quadrant arc VQ = π/₄ (in a linear sense, as opposed to angular measurement in radians).

In this case the VD height of the triangle will be VC+CD ie
0,5 + Φ³/₂ [0,2360679774997896964/2… = 0,1180339887498948482…+ 0,5], which in our construct turns out to be the golden section Φ of the diameter = 1:
0,618033988749894848204586834365638117720309179805762862135…

To measure VA it will be neessary according to the theorem of Pythagoras:
AD = √AC² – CD²