Triangles Touching Three Sides of a Square

We study triangles whose three vertices lie on three different sides of a square. For a fixed triangle, we derive an exact criterion for such a placement and prove that the admissible square side lengths form a single interval. After normalizing the longest side of the triangle, we determine the smallest admissible square explicitly by a four-region geometric classification. As an application, we count congruence classes of integer-sided triangles with this property; the resulting sequence is OEIS A400509. We prove that a(n) = C n³ + O(n²), where C = 0.0776924549... is given by an explicit two-dimensional integral.