Integer-Sided Triangles Meeting Three Sides of a Square

We study integer-sided triangles whose vertices can be placed on three distinct sides of a square, one vertex on each side. We give an exact criterion for such placements, prove that for every fixed triangle the admissible square side lengths form an interval, and determine the minimum square size by a four-region classification of triangle shapes. We then count the resulting congruence classes and prove the asymptotic formula a(n) = C n^3 + B n^2 + o(n^2), with C = 0.0776924549474... and B = 0.287883924575841.... The associated sequence is OEIS A400509. The record includes the manuscript, LaTeX source, verification code, figure source, and a b-file for n = 1, ..., 400.