I recently discovered a formula, my proof is really a high school proof in three lines.

$$4\sum_{x, \, y \, \in \, \mathbb Z_{\geq 0}^2, \, \det(x \ \ y) = 1} \frac{1}{\lVert x\rVert^2\cdot\lVert y\rVert^2\cdot \lVert x+y\rVert^2} = \pi$$

We sum over all pairs of lattice vectors in the first quadrant such that the oriented area of the parallelogram spanned by them is one. Note that each primitive vector in the first quadrant appears as $x+y$ for some unique pair $x$ and $y.$ So it can be seen as a kind of zeta function for Gaussian integers.

It should be known to humanity but I cannot find anything similar. Who can suggest a source where to look for such formulae?

I will later post my proof here but before I encourage you to suggest another proofs =) Maybe this way we can discover some nice connections!