I think centroid and center of mass are synonymous for a flat uniform plate? I am used to "barycenter" as used in astronomy, the center of mass of two distant celestial objects.

Yes, for a uniform object they are the same thing. Bary-center literally means (from Greek) weight-center. When I was in school (in Spanish) we called the centroid of a triangle the barycenter.

Hopefully ta0 is sending you that equation you are looking for (maybe a snapshot from his notes?).

I added a couple of dimensions to the drawing above.

If you equate the lever of both figures, you get:

1 m = (x -1) m x

where m is the mass of the small line figure, and the ratio of dimensions is x. From there the equation x

^{2} - x - 1 = 0, which gives x equal to the Golden Ratio.

If you were attaching two planar figures, the equation would be:

1 m = (x -1) m x

^{2}as ortwin pointed out. The solution would be x = 1.46557