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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:
  1. 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.
  2. 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…
  3. 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