Does the triangle inequality apply?

Let $a$ be the smallest side. Then it will follow that the other sides are $ar$ and $ar^{2}$ for some positive number $r$ .

We'll have to make sure that the sum $a( 1 + r + r^2)$ is an integer. Also, we'll have to make sure that $a$ , $ar$ and $ar^2$ are all integers. Finally, we'll have to make sure that $a + ar > ar^2$ , or $1 + r > r^2$ .

obviously, $1 < r < 2$ , as the inequality above fails to hold otherwise. So we select the rational number between with the lowest possible denominator, so that $1 + r + r^2$ can be multiplied with the least integer possible. With that we get $r = \frac {3}{2}$ .

Now, substituting $r = \frac {3}{2}$ , we have to set $a$ such that $a$ , $ar$ , and $ar^2$ are integers. By inspection we see that the least possible value for $a$ would be $4$ .

And with that the total perimeter is $4 (1 + \frac {3}{2} + (\frac {3}{2})^2) = 19.$

Don't forget to reshare! :)