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Geometric Triangles

 Published on Sunday, 5th February 2012, 07:00 am; Solved by 560;
Difficulty rating: 65%

Problem 370

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

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

There are $861805$ geometric triangles with perimeter $\le 10^6$.

How many geometric triangles exist with perimeter $\le 2.5 \cdot 10^{13}$?