You wrote
> Question: Does anyone know more such "dramatically simple" (to quote
> David F.) constructions of the phi in the equilateral triangles?
There is my 3 circle theorem:
Take 3 concentric circles with radii in the ratio of 4 : 2 : 1.
Draw a tangent to the inner circle that meets the middle circle
at A and C.
Extend the tangent line from C outward to meet the outer circle at B.
I say that the line AB is cut in a golden section at C.
Furthermore, as pointed out to me by Fishback, there is a golden section
produced not only when the radii are in the ratio of 4 : 2 : 1, but also
when, with the radii are called c, b, a, they solve the diophantine equation
c^2 = 5b^2 - 4a^2.
This diophantine equation has infinitely many solutions.
Best wishes,
Sam Kutler