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 Φ^{3};
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 Φ^{3} in the middle of AB ; in particular a line parallel to AB from the point X is tangent to Φ^{2}, 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 indepth 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
