Problem 370: Geometric triangles

Let us define a geometric triangle as an integer sided triangle with sides $a ≤ b ≤ c$ so that its sides form a geometric progression, i.e. $b^2 = a \times c$.

An example of such a geometric triangle is the triangle with sides $a = 144$, $b = 156$ and $c = 169$.

There are $861\,805$ geometric triangles with $\text{perimeter} ≤ {10}^6$.

How many geometric triangles exist with $\text{perimeter} ≤ 2.5 \times {10}^{13}$?