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 \[\frac{1}{2} \lt r \lt 2\]
.