This is indeed an old trick, though there's nothing "magic" about it as
the link says.

Also, the link says that the "long compass" thus constructed is
universally applicable to arcs of a particular radius. That's not
true. It could be used to draw arcs of any radius, but the arc length
subtended will stay the same. In other words, when you construct your
original three points, let's say you put brads at 10:00, 12:00, and
2:00 on the perimeter of a 15' diameter clock. The angle between the
outer two points would be 4/12 of 360 degrees = 120 degrees. At that
point, in order to trace out the 120 degree arc, you'd need to remove
the middle brad and trace the path of the crotch as you rotate the
"compass" from one brad to the other. For that matter, you could take
ANY two brads any distance apart and use the compass to trace out a 120
degree arc between them. The radius would not be fixed; it would be
whatever radius is required in order for those two brads to be 120
degrees apart.

If you want to know the "maths", the radius can be found as follows:

R = D/(2*sin(arc/2)), where D is the distance between the two outer
brads, and arc is the arc angle (120 degrees in this case).

Nuances notwithstanding, this would be a pretty good trick to use with
a router.

Josh