I agree that proof was not clear. It was more a case of "if it's the Golden Ratio it works", than deriving the Golden Ratio.

Let me try again with an even more general result.

1. The center of mass (centroid) of the full figure has to be in the segment between the center of mass of the hollowed figure and the center of mass of the hole.

2. For the calculations you can consider the masses (i.e. areas) of the figures concentrated in their centroids, so they can be treated as point masses. Therefore, the shape and orientation of the figures doesn't matter.

3. Let the area of the hole be A/k^{2}, where A is the area of the full figure.

4. Let the distance between the centroid of the hollowed figure and the centroid of the full figure be: a

5. Let the distance between the centroid of the hole and the centroid of the full figure be: a x

Equating the levers at the centroid of the full figure:

(A-A/k^{2}) a = A/k^{2} a x => k^{2} - 1 = x

So, for an arbitrary hole, the new centroid will be in line with the centroid of the original figure and the centroid of the hole, on the opposite side of the later, at a distance closer by a factor of 1/x = 1/(k^{2}-1)

6. If k = φ (Golden Ration), x = φ^{2} - 1 = φ