Thought I would write again to let you know I followed through and verified that phi is *not* the ratio of R/r nor of R/(R-r) in the problem I originally presented. So the arc of 144 degrees is *not* related to the arc of 60 degrees by a chord on two concentric circles, when the respective radii are related to by the golden ratio.
However, I decided to check and see what larger arc *was*, if i stipulated the 60-degree arc on the smaller circle and the radii being in that ratio.
Let there be two circles with a common center O. A chord subtends an arc of 60 degrees on the smaller circle, and a to-be-determined arc on the larger. Let h be the distance from O to this chord; it makes a right angle with the chord at point H, slightly inside the circumference of the smaller circle.
The radius r of the smaller circle is 1; The distance from the circumference of the smaller circle to the larger is phi. So the radius R of the larger circle = 1+phi.
There are straight lines connecting the center O to the intersection of the chord to the circumferences of the two circles. On the smaller circle the point of this intersection (call it A) is at a 60 degree angle. On the larger circle the point (call it B) is at an angle to be determined.
Since r=1, sin A= sin60= h/1 = h. So h= sin60 = ((sqrt3)/2)
sin B = ((Sqrt3)/2) / (1 + phi)
= (sqrt3)/(2(1 + phi))
so B = arcsin (sqrt3)/(2(1 + phi))
= approx 19.31687 degrees
(I confess I turned to an arcsin calculator, and used a long decimal approximation of phi). Now I could easily calculate the remaining angle of the other angle in triangle HOB. The I subtracted 30 degrees to get angle AOB.
Multiply AOB by 2, and add 60 degrees, and voila!
(naturally, I could also have just doubled HOB)
It isn't 144, but much closer to the 141 or so which I kept getting when I had tried with compass and straightedge and measuring with a protractor. Which makes me feel a little better about my steadiness with good old Euclidean tools.
Thanks again for your help, without which I would not have got started.