n - gon With Sides in Geometric Progressions

How many polygons may have sides in a geometric progression?
Suppose we have a triangle. The sides are  
\[a, \: ar, \: ar^2\]
. The longest side must be less than the sum of the two shorter sides, so

\[a \lt ar+ar^2\]
,  
\[r \lt 1\]

\[a + ar \lt ar^2\]
,  
\[r \lt 1\]

Cancelling  
\[a\]
  and solving both inequalities gives  
\[\frac{\sqrt{2}-1}{2} \lt r \lt \frac{\sqrt{2}+1}{2}\]

We can perform the same analysis for each n - gon. Suppose the n gon has an infinite number of sides. Then
\[a \lt ar+ar^2+ ...+ ar^{n-1}+...\]
,  
\[r \lt 1\]

\[a + ar+...+ar^{n-1}+... \lt ar^2\]
,  
\[r \lt 1\]

Cacelling  
\[a\]
  as before and solving gives  
\[\frac{1}{2} \lt r \lt 2\]
.