Two Triangles from Neighboring Squares: Geometric Rearrangement, a Shared Leg, and Consecutive Hypotenuses

Two integer right triangles can share a leg while their hypotenuses differ by exactly one. This paper approaches such pairs through neighboring squares: different corner squares are removed, leaving the same square area. An explicit dissection follows the retained regions into rectangles and then into a common square. The opening example connects the triangles (7,24,25) and (10,24,26) through twelve identical rectangular tiles. For positive unequal integers x and y, the construction uses N = 2xy + x + y and A = 4xy(x + 1)(y + 1). Whenever A is a square, it supplies the common squared leg; conversely, every positive integer triangle pair with a common leg and consecutive hypotenuses returns to this description. Infinite families arise by matching the square-free parts of consecutive-integer products. Holding the common leg fixed instead gives a finite divisor table, which identifies 24 as the smallest possible shared leg. The paper also distinguishes shape from scale. A finite factor test determines which integer scales of a fixed triangle shape can participate in a pair with hypotenuse gap one. Allowing larger gaps reveals why that one-unit condition is restrictive. The purpose is understanding through geometric presentation and connections among established identities, rather than a claim of historical novelty. Developed through Human-AI collaboration with Bill Widi as curator. ChatGPT assisted with deductions, algebraic formulation, exact computational checks, geometric dissections and illustrations, drafting, and editing. Bill Widi directed the investigation, reviewed the exposition, and guided the revisions.