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

S M Nazmuz Sakib's Tangent-Length Law for Triangle Angles | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 13 August 2025 V1 Latest version Share on S M Nazmuz Sakib's Tangent-Length Law for Triangle Angles Authors : Nazmuz Sakib and S M Nazmuz Sakib 0000-0001-9310-3014 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.175510718.82689471/v1 160 views 134 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract 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 () 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. Supplementary Material File (manuscript2.pdf) Download 347.52 KB Information & Authors Information Version history V1 Version 1 13 August 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords acute triangle test incircle tangent lengths law of cosines obtuse triangle test s m nazmuz sakib triangle geometry Authors Affiliations Nazmuz Sakib School of Business And Trade, ) Graduate of BSc in Business Studies Faculty of Law, Dhaka International University, ) Graduate of LLB (Hon's) Graduate of BEng in IT, Harris University College of Liberal Arts and Sciences, ) Graduate of MBA (Human Resources, International MBA Institute Scholars Academic and Scientific Society Professor of Science in Research And Development, Charter University View all articles by this author S M Nazmuz Sakib 0000-0001-9310-3014 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 160 views 134 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Nazmuz Sakib, S M Nazmuz Sakib. S M Nazmuz Sakib's Tangent-Length Law for Triangle Angles. Authorea . 13 August 2025. DOI: https://doi.org/10.22541/au.175510718.82689471/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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