While looking at the Wikipedia entry for the Supergolden ratio aka Narayana's cow constant (cf. https://en.wikipedia.org/wiki/Supergolden_ratio), I noticed the following construction involving that constant and a 120-degree ($2\pi/3$ radian) triangle on that page:
I then began to wonder about similar constructions with other fractional angles: $2\pi/2$, $2\pi/4$, ...
I found that an angle of $2\pi/2$ gives a degenerate triangle based on the golden ratio:
$2\pi/4$ gives a right triangle based on the square-root of the golden ratio:
And $2\pi/6$ gives an equilateral triangle based on unity:
Nothing too surprising or new in the results so far, but an angle of $2\pi/5$ (72-degrees) produced the following:
So, the positive solution to $-2 - x + \sqrt 5 x - 2 x^2 + 2 x^4=0$, approximate value $1.11716..$, minimal polynomial $1 + x + x^2 + x^3 - x^4 - x^5 - 2 x^6 + x^8$
Is this a known/named constant?