The following figure points out that like the circle 3 fills the dis­tance from below the triangle’s base, to bottom of the outer circle 1, the circle 2 does the same on to the other side from above the base, which shows us with ease that the height of the Triangle is even in Phi ratio with the outer circle diameter!

This revealed an alternative way to draw the complete golden triangle - all its sides being involved - and instructs us of many interesting detais con­cern­ing the relations between our ‘el­e­men­ta­ry’ rec­tan­gle and the con­cen­tric circles scaled in Golden Ratio, which look like to have beeen pre­sent behind the scenes at each step of our process, giving rise to what we can settle from now on as the “Golden Circles Ratio.

The Kepler's cite “division of a line into extreme and mean ratio” it was thus o­ver­come,
as not only linear.
In deed a golden expansion map
is defined by a se­quence of con­cen­tric waves, whose di­am­e­ters are reg­u­lat­ed by increase of the golden ratio j power – tap the figure if HTML.

The next step dis­closes one still un­known gold­en spi­ral, which works out through this plan, ie. cros­sing eve­ry cir­cle at the same [zero] degree.
Our Great Triangle handles all these and much more, like a gate be­tween the plain for­mu­las and the revolving energy!

Squaring of the Circle?

Yes, provided the disclosure of the geometrical π !
On July 2021, right after the inspiration of the Golden Cosmic Egg on the next page in­ser­tion – which appears to me to have been introductory – something more led me to analyze the ir­re­solv­a­ble problem of the Squaring of the Circle with ruler and com­pass. Despite the claims of im­pos­si­bil­i­ty world­wide, the solution was ready there, a­gain in that a­maz­ing triangle with the precious mas­tery of its golden sections.
Although It could be condensed here, this would be a poor presentation compared to what it de­serves, and the space wouldn't be enough an­y­way. Another domain like this one was ded­i­cat­ed to explain the natural π discovery, and a portable 33 pages [legal format] PDF free ebook was pub­lished «2×2=3.14» illustrating the de­tails [ISBN 978-88-904390-5-6].
Furthermore other domains added their con­tri­bu­tion, but so far all is in Italian language only; ex­cept a special curious genealogical exposition con­cern­ing the topic, at AstroTime, which not only erec­ted and analyzed the astral chart referring to the recurrence of the ‘pi day’, but also dem­on­strat­ed, geometrically of course, the π's genesis with even maternity and paternity. (figures to the right)


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