Analytical Conditions of Existence for Right Triangles: AnIsoperimetric Approach

Abstract

This article investigates the morphological reconstruction of right triangles based exclusively on their area (A) and perimeter (p). An algebraic modeling based on the Pythagorean theorem  generates a parametric system that defines the side lengths. Discriminant analysis reveals that the existence condition imposes a severe constraint on the isoperimetric quotient (K = p^2 / A). It is demonstrated that the isoperimetric limit shifts from the universal boundary 12 * raiz(3) to the strict restriction K >= 12 + 8 * raiz(2), corresponding to the isosceles right triangle. The parametric space is partitioned, visually mapping the orthogonality zones. Additionally, the model is applied to Number Theory, defining divisibility criteria for the formation of Pythagorean triples.