A common way to construct the golden ratio is to construct two
perpendicular segments joined at endpoints,
whose lengths are in the ratio $2:1$
like the segments $AC$ and $AD$ in the figure in the question.
One then constructs the segment connecting the other two ends of
these segments to form a right triangle
(whose hypotenuse is $CD$ in the figure in the question).
Finally, one marks off the length of the shorter leg of the triangle
($AD$ in the question, but labeled $BC$ in the figure below)
on the hypotenuse (at point $E$ in the question, $D$ in the figure below).
The remaining portion of the hypotenuse
($DE$ in the question, $AD$ in the figure below) has length
$\sqrt5 - 1$ times the length of the shorter leg of the triangle,
and therefore is in the ratio $\phi : 1$ (the golden ratio) with
the longer leg.
There are examples of this construction at
A typical diagram of the construction looks like the following
figure, taken from
There are a number of ways you can construct the two perpendicular
segments, of which putting two congruent squares side by side
(as in New Golden Ratio Construction with Two Adjacent Squares and Circle. Have you seen anything similar?)
or by setting two congruent tangent circles' centers on a line to
mark the longer leg, and use one of the circles to mark the distance
on the shorter perpendicular leg.
The triangles in the figure in the question and the
figure from Wikimedia are differently oriented (reflected left-right)
and labeled with different letters, but those are unimportant differences.
The main difference between the figures is that in the Wikimedia figure,
the length of the shorter leg is marked off from the end of the
hypotenuse on the shorter leg, whereas in the question that length
is marked off from the end on the longer leg.
If the two legs of the triangle are formed from the edges of two
adjacent squares, as they are in New Golden Ratio Construction with Two Adjacent Squares and Circle. Have you seen anything similar?, then even this difference is not as significant,
since we have two copies of the triangle.
Note: At this point in the prior-art construction
we have already constructed segment $AD$ so that
$AB : AD$ is the golden ratio. If $BC = 3$ (the radius of the circles in
the question), then $AD$ in the prior-art construction is exactly congruent
to $DE$ in the question.
If we must have both segments in the ratio lie end-to-end along
the hypotenuse of the triangle, we could simply complete the circle
with center at $C$ and extend segment $AC$ to meet the circle at $F$,
at which point $DF : AD$ is the golden ratio,
and the three-point figure $ADF$ in this construction would be congruent
to the figure $DEF$ in the question.
Additional Note: This paragraph is not particularly
relevant to the question, except that it shows a slight advantage to
doing the construction in the "Wikimedia" fashion: namely, after having
already constructed the golden ratio once,
we get a "bonus" of constructing a second pair of segments in the golden
ratio with merely one action of a collapsible compass.
Namely, we strike an arc with center $A$ from $D$ to $E$,
thereby copying the length of $AD$ onto the long leg of the triangle
dividing the segment $AB$ into
the two parts $AE$ and $EB$, which also are in the ratio $\phi : 1$
(the golden ratio).
To divide the segment $AC$ in the question in this fashion, we have to
construct a point $P$ between $A$ and $C$ such that $CP \cong DE$.
This is easy enough with a non-collapsing compass (use segment $DE$ to
set the compass, then put one end of the compass at $C$ and strike an
arc across the segment $AC$) but rather tedious to do with a
collapsible compass (although that too is a standard classical