Geometric Triangles

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 $\le 2.5 \cdot 10^{13}$?

This mathematics problem appears to involve number theory and geometric sequences, and it requires the use of programming to solve. One efficient approach involves generating Pythagorean triples.

A couple of vital facts for this problem:

1. Every Pythagorean triple with ‘a’ and ‘b’ as legs and ‘c’ as hypotenuse takes the form:
a = k*(m^2 – n^2)
b = k*(2*m*n)
c = k*(m^2 + n^2)
It is significant because it directly relates the problem to Pythagorean triples as b^2 = a*c turns out to correspond to Pythagorean triples where one leg length and hypotenuse are perfect squares.

2. Any primitive Pythagorean triple (a triple in which ‘a’, ‘b’, and ‘c’ are coprime) can be expressed as:
a = m^2 – n^2
b = 2mn
c = m^2 + n^2
With m > n > 0, m and n are coprime and not both odd.

Now, let’s solve the problem. Using the association with Pythagorean triples, we define a = x^2, b = y, and c = z^2 where x^2*z^2 = y^2 satisfies the conditions to form a geometric triangle. These correspond to the Pythagorean triangle properties above which gives us an easier way to enumerate the number of solutions.

The most significant challenge is finding the upper bound for m such that the perimeter is less than 2.5 * 10^13. This value turns out to be roughly 5 * 10^6. With this value, we can create a nested loop (the outer loop iterating through all possible ‘m’ and the inner loop all possible ‘n’ for each ‘m’) counting how many Pythagorean triples can be made that satisfy the conditions.

However, this implementation considers only primitive triangles. If a triangle is not primitive, it can be scaled down to a primitive triangle by dividing a, b, c by their greatest common divisor. So, we must also count these non-primitive triangles. We multiply by a factor related to the square root of the triangle’s perimeter, carefully handling rounding and subtracting one to exclude the primitive triangle itself.

This approach requires careful development and optimization of a computing algorithm and probably wouldn’t be practically solvable manually.

Please note that for implementing this solution, good knowledge of a programming language is needed along with an understanding of number theory. The final solution for how many geometric triangles exist with perimeter ≤ 2.5 * 10^13 is not a straightforward number and would be the output of the algorithm.

This is a broad outline of the solution and may vary depending on specific implementation details.

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