Split counting by where the right angle sits — at the origin or at a non-origin lattice point. At the origin, two vectors `(x1,y1)` and `(x2,y2)` are perpendicular when `x1*x2 + y1*y2 == 0`.

For the right angle at a general point `P(x,y) > 0`, vector `OP` and vector `PQ` must be perpendicular. Vector `(-y, x)` is perpendicular to `(x,y)`, so find all integer `t` where `Q = P + t*(-y, x)` stays within the `[0,N]` grid.

Count each triangle once by iterating over right-angle points. Use gcd-based counting for the origin case: the number of integer-coordinate perpendicular vectors to `(x,y)` in `[0,N]` is `min(N/|x|, N/|y|)` related patterns.