A023022 and an observation about random triangles in polygons

[Eric W. Weisstein]

Find the average area of a triangle picked at random in a regular n-gon
(n>=3) of unit area.  Rather amazingly, this seems to be an algebraic
number of order precisely phi(n)/2 (i.e.,
http://www.research.att.com/projects/OEIS?Anum=A023022).

Can anyone see a geometric reason why this should be the case (other than
resorting to some algebra on the closed-form formula due to Alikoski 1939;
http://mathworld.wolfram.com/PolygonTrianglePicking.html)?

An immediate result is that only the triangle, square, and hexagon give 
rational average areas.

Cheers,
-Eric