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## The KEY Once the four circles have been identified by their respective diameters in decreasing gold¡en pro¡por¡tion, let's now move on to the first initiative, which con¡sists in tracing from the vertical point `V` on the larg¡er cir¡cum¡fer¡ence, of diameter = `1`, a line tan¡gent to the circumference Φ, which will meet the primary at point `A`. With or without al¡ge¡bra¡ic proof – I pro¡duced it here in 2009 – we will im¡me¡di¡ate¡ly 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 fol¡low¡ing 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 comeà
3. The sum of `AV` and `VB` is equivalent to `AB×`Φ, i.e. the base of this es¡pe¡cial tri¡an¡gle is the Golden Section of the sum of the sides; the supreme bal¡anc¡ing 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 de¡ci¡sive; and now the time has come to grap¡ple with the num¡bers.

Given the diameter of the pri¡ma¡ry circle as a unit of meas¡ure¡ment, therefore with a val¡ue of `1.0`, the cir¡cum¡fer¡ence will be `= π` and the quad¡rant arc `VQ = π/4` (in a linear sense, as opposed to angular meas¡ure¡ment in radians).

In this case the `VD` height of the tri¡an¡gle 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 = √AC▓ – CD▓`