The Baran Quadrilateral Constraint: Diagonal Elimination and Sign-Free Formulation

This paper presents an original result in elementary geometry, independently conceived and formulated by Abdullah Baran. A quadrilateral is divided by a diagonal into two triangles with outer sides a, x and c, y and areas A1, A2. The diagonal b is eliminated to yield a single constraint among the six quantities a, x, c, y, A1, A2 alone. The key factored form of the radical is sqrt((ax - 2A1)(ax + 2A1)), which makes the radicand manifestly non-negative and the expression easy to compute. The paper further proves that the sign ambiguity (the plus-or-minus in the constraint) is removed entirely by a condition that references only the given data, with no mention of b: if a2 + x2 > 2·sqrt((ax - 2A1)(ax + 2A1)) and c2 + y2 > 2·sqrt((cy - 2A2)(cy + 2A2)), then both included angles are acute, both signs are fixed at +1, and the constraint takes a unique, unambiguous form. **Keywords:** cosine rule, triangle area, quadrilateral, diagonal elimination, sign elimination, Baran quadrilateral constraint, elementary geometry License: CC BY 4.0 Author: Abdullah Baran — [email protected] — x.com/realABaran