S M Nazmuz Sakib's Tangent-Length Law for Triangle Angles

A novel geometric criterion, presented which determines whether a triangle’s angle is acute, right, or obtuse using only the tangent lengths from its incircle. Let x A =s−a, x B =s−b, x C =s−c be the equal tangent lengths from vertices A,B,C to the respective incircle contact points, where a,b,c are the triangle’s sides and s is its semiperimeter. The identity x A x B [?]sx C classifies ∠C as acute (<), right (=), or obtuse (>) without directly measuring angles or using trigonometric functions. This result follows from a re-expression of the law of cosines in tangent-length form and applies cyclically to all angles. Despite the simplicity of its proof, no prior documentation of this exact formulation appears in mainstream mathematical literature, suggesting that it is a new, elegant tool for elementary geometry and olympiad-style problem solving.