Let us assume that there exists a triangle with measures of its angles in a Geometric Progression (G.P.) with a common ratio other than 1.
Then what are the possible ranges of (that is starting set and ending set) of measures of angles - when calculated to 2 decimal places of precision?
Is there a point of inflexion / break-even in this case? (By this I mean - one set of solutions become complement of the other set - thereby no need for us to find by calculating but we can infer by inspection) If so, how to find it?
The general form for three numbers in geometric progression is $a, ar, ar^2$. Here of course $a > 0$. We can assume $r \geq 1$, or else the angles are just in decreasing rather than increasing order. We know the sum of the angles must be $\pi$ radians or $180^\circ$.
None of these restrictions put any upper bound on $r$. So choosing any $r \geq 1$, there's a triangle with angles
$$ \frac{180^\circ}{r^2+r+1}, \frac{180^\circ r}{r^2+r+1}, \frac{180^\circ r^2}{r^2+r+1} $$
If for example $r=1000$, the angles can be approximately $0.00017982^\circ, 0.17982^\circ, 179.82^\circ$, close to a degenerate linear triangle. Of course the $r=1$ limit gives the equiangular $60^\circ, 60^\circ, 60^\circ$.