S M Nazmuz Sakib's Theorem of Symmetric Perpendicular Bisectors in Isosceles Triangles

Prof. Engr. Dr. S M Nazmuz Sakib

doi:10.33367/jtme.v3i2.7785

Authors

Prof. Engr. Dr. S M Nazmuz Sakib Dhaka International University; School of Business And Trade; International MBA Institute; Atlanta College of Liberal Arts and Sciences https://orcid.org/0000-0001-9310-3014

DOI:

https://doi.org/10.33367/jtme.v3i2.7785

Keywords:

Isosceles Triangle, Perpendicular Bisector, Symmetry, Geometric Theorem, Segment Sums

Abstract

This paper introduces a new geometric theorem focused on isosceles triangles, specifically examining the properties of perpendicular bisectors. The theorem asserts that in an isosceles triangle, where two sides are equal, the perpendicular bisectors of these sides intersect at a point such that the sum of the lengths of the segments on one bisector is equal to the sum of the lengths of the segments on the other bisector. The theorem is validated through a quantitative research methodology involving geometric constructions, where perpendicular bisectors are drawn for selected isosceles triangles. Mathematical calculations are employed to measure the lengths of the segments formed by the intersection of the bisectors. These segment lengths are then compared and analyzed across various configurations of isosceles triangles, including triangles with different orientations and base lengths. Case studies are conducted to test the theorem under different geometric conditions, ensuring the results are consistent. The methodology is further supported by mathematical proofs, which are derived to formally validate the relationship between the segment sums on each perpendicular bisector. This discovery provides a novel insight into the symmetry of isosceles triangles and contributes to the broader understanding of geometric properties in symmetrical shapes.

Author Biography

Prof. Engr. Dr. S M Nazmuz Sakib, Dhaka International University; School of Business And Trade; International MBA Institute; Atlanta College of Liberal Arts and Sciences

Prof. Engr. Dr. S M Nazmuz Sakib, CMSA®, FPWMP®, FTIP®, BIDA®, FMVA®, CBCA® is a Fellow at the Scholars Academic and Scientific Society, Member of International Association of Engineers - IAENG, a Bangladeshi Arena Grand Master (AGM) in FIDE (International Chess Federation), and a lifetime member of the Bangladesh English Language Teachers Association (BELTA). His educational journey began with the Primary School Certificate (PSC) from Cotton Research Station School in Sreepur, Gazipur, after which he studied Class 6 at Alhaz Dhanai Bepari Memorial (ADBM) High School in Sreepur before relocating to Jashore. He completed his Junior School Certificate (JSC) from Jagodishpur Mirzapur Ismail Secondary School in Jagodishpur, Chowgacha, Jashore, and his Secondary School Certificate (SSC) from A.K. High School and College, Dhania, Dhaka. He obtained his Higher Secondary Certificate (HSC) from Chowgacha Government College in Chowgacha, Jashore.
Engr. Sakib holds a Bachelor of Science in Business Studies (BSc) from the School of Business and Trade, an online business school in Switzerland, awarded in 2021 with a CGPA of 4.0 out of 4.0 (1st Class, 1st) and 97.06% marks. He also completed his Master of Business Administration (MBA) in Human Resources in 2023 from the International MBA Institute, an online business school in Switzerland. In addition, Dr. Sakib pursued a Bachelor of Laws (Honors) (LLB) at Dhaka International University, Dhaka, Bangladesh, graduating in 2023 with a CGPA of 3.31 out of 4, and a Bachelor of Engineering in Information Technology from Harris University in Liverpool, UK, in 2024 with a CGPA of 3.75 out of 4. He earned his Doctor of Business Administration (DBA) from Atlanta College of Liberal Arts and Sciences in 2025, graduating with A+ (Distinction).
Dr. Sakib got various Professional Certifications i.e. Unilever Digital Marketing Analyst, Unilever; AWS Cloud Solutions Architect, Amazon Web Services; Microsoft UX Design, Microsoft; Liberty Mutual Insurance Sales Agent, Liberty Mutual Insurance; Microsoft UX Design, Microsoft; Keller Williams Real Estate Agent, Keller Williams Realty, Inc.; HRCI Human Resource Associate, HRCI; SAS Statistical Business Analyst, SAS Education; Google Project Management, Google; Google Business Intelligence, Google; Google Digital Marketing and E-Commerce, Google; Google IT Support, Google; Google Data Analytics, Google; SAS Visual Business Analytics, SAS Education; Meta Social Media Marketing, Meta; Tally Bookkeeper, Tally BookkeeperTally Education Private Limited (TEPL); Strategy and Finance for the lifecycle of a Social Business, Yunus Social Business; Sustainability Analyst Fundamentals, Arizona State University; PepsiCo: Water Stewardship, PepsiCo; Business and Marketing Strategies, University of London; Data Exploration in Psychology, American Psychological Association & PsycLearn; American Politics, Foreign Policy, and US-India Ties, O.P. Jindal Global University; Modern and Contemporary Art and Design, MoMA The Museum of Modern Art; Digital Health, Imperial College London; Advanced Digital Transformation, IIM Ahmedabad; Qualitative Research Design and Methods for Public Health, Emory University; Qualitative Research Methods in Psychology, American Psychological Association & PsycLearn; Psychological Research, American Psychological Association & APA PsycLearn; Energy Production, Distribution & Safety, University at Buffalo & Stony Brook University; Project Management, University of Colorado Boulder; Introduction to Blockchain, American Institute of Certified Public Accountants (AICPA); Graphic Design, California Institute of the Arts; Managing Major Engineering Projects, University of Leeds; Recommender Systems, University of Minnesota; Six Sigma Yellow Belt, Kennesaw State University; AWS Fundamentals, Amazon Web Services; Engineering Project Management, Rice University; Autodesk Generative Design for Manufacturing, Autodesk; Project Management Principles and Practices, University of California, Irvine; Project Management & Other Tools for Career Development, University of California, Irvine; Six Sigma Green Belt, Kennesaw State University; Career Success, University of California, Irvine; Human Resource Management: HR for People Managers, University of Minnesota; Autodesk CAD/CAM/CAE for Mechanical Engineering, Autodesk.
Engr. Sakib's CPD Diploma Certifications include Advanced Diploma in Political Ideologies; Advanced Diploma in Tissue Engineering; Advanced Diploma in Genetic Engineering - Theory and Application; Diploma in Fashion Design; Diploma in Nutrition, Therapeutics and Health; Diploma in Biology; Diploma in Pharmacy Technician; Advanced Diploma in Soil Science and Technology; Diploma in Energy Economics, Energy Systems and Environmental Impact; Diploma in Community Psychology; Diploma in Training of Trainers; Advance Diploma in Principles of Industrial Engineering; Advanced Diploma in Production & Operation Management; Advanced Diploma in Modelling and Analytics for Supply Chain Management; Diploma in Effective Human Resource Administration; Diploma in Audio System Engineering; Diploma in ISO Standards, specialized - Integrated Management System (IMS); Diploma in Lean Manufacturing - Productive Management with Fundamental Tools; Advanced Diploma in Tourism and Hospitality Management; Diploma in Web Design; Diploma in Human Resources; and Diploma in Basic English Grammar.
Dr. Sakib's publications are A Novel Approach for Multi-cluster-Based River Flood Early Warning System Using Fuzzy-Logic-Based Learning and Rule Optimization, Applications of Fuzzy Logic in Decision Making and Management Science, Springer Nature; Group Revision is Better Than Self-Revision in Case of Mathematics, Vol. 3 No. 1 (2025): Noumerico, March 2025, Noumerico: Journal of Technology in Mathematics Education, Mathematics Education Department, Tribakti Lirboyo Islamic University; The 2003 US Intervention of Iraq: Objectives, Implications, and Global Security Dynamics, Handbook of Migration, International Relations and Security in Asia, Springer Nature; The Detrimental Impacts of Deforestation: Causes, Effects, and Potential Solutions, VOLUME: 06 ISSUE: 02, Journal of Natural and Applied Sciences of Pakistan, Kinnaird College for Women, Pakistan; Optimizing Beneficial Oral Hygiene Care: Transitioning from Manual Brushing and Utilizing Powered Toothbrushes to Improve Plaque Control and Prevent Gingival Inflammation, Update Dental College Journal, Update Dental College (UpDC), BD; Blockchain Technology for Smart Contracts, Blockchain Technology in the Automotive Industry, CRC Press, Taylor & Francis Group; Kinetics of Sodium Hydroxide and Ethyl Acetate Reaction in A Continuous Stirred Tank Reactor: A Comparison of Experimental and Theoretical Conversion, Volume 6 Issue 1 of Journal of Natural and Applied Sciences of Pakistan, Kinnaird College for Women, Pakistan; Evaluation of Three-Dimensional Reconstruction Technology in Precision Hepatectomy for Primary Liver Cancer, Formosan Journal of Surgery, Taiwan Surgical Association & Wolters Kluwer Health, Inc.; Internet of Medical Things (IoMT) for Remote Healthcare Monitoring Using Wearable Sensors, Vol 4 No 2 issue of International Journal of Computing and Related Technologies (IJCRT), Sindh Madressatul Islam University; Analysis of Fundamental Algebraic Concepts and Information Security System, Vol. 2 No. 1 (2024) of Noumerico: Journal of Technology in Mathematics Education, Mathematics Education Department, Tribakti Lirboyo Islamic University; Mathematical Models and Formulas for Language Development and Disorders: A Collection of Hypothetical and Novel Approaches, Childhood Developmental Language Disorders: Role of Inclusion, Families, and Professionals, IGI Global Scientific Publishing; Blockchain Technology for Smart Contracts: Enhancing Trust, Transparency, and Efficiency in Supply Chain Management, Achieving Secure and Transparent Supply Chains With Blockchain Technology, IGI Global Scientific Publishing; Salutogenic Marketing in the Elderly: Leveraging Digital Transformation in Geriatric Dentistry - Creating Positive and Meaningful Experiences for Older Adults, Geriatric Dentistry in the Age of Digital Technology, IGI Global Scientific Publishing; Fixed Point Theory and Insurance Loss Modeling: An Unlikely Pairing, Advancement in Business Analytics Tools for Higher Financial Performance, IGI Global Scientific Publishing; Navigating the New Frontier of Finance, Art, and Marketing: A Look at Cryptocurrencies, NFTs, and Metaverse, Concepts, Technologies, Challenges, and the Future of Web 3, IGI Global Scientific Publishing; Assessing enrichment and contamination of sediments in the effluent canal of the ore processing industry and Naviundu River in Lubumbashi, Democratic Republic of Congo, Vol. 58 (2023) of EQA (International Journal of Environmental Quality), Department of Agricultural and Food Sciences – Alma Mater Studiorum – University of Bologna; The Role of Innovation in Driving the Bioeconomy: The Challenges and Opportunities, Handbook of Research on Bioeconomy and Economic Ecosystems, IGI Global Scientific Publishing; Exploring the Intersection of Software Engineering and Mobile Technology from 2010 to 2021: A Review of Recent Research, Journal of Innovation Information Technology and Application (JINITA), Pusat Penelitian dan Pengabdian Masyarakat (P3M), Politeknik Negeri Cilacap (Cilacap State Polytechnic), Indonesia; Comparing the sociology of culture in Bangladesh and India: Similarities and differences in Bangladeshi and Indian cultures, Volume 6, Issue 1, June 2023, Simulacra, Center for Sociological Studies and Community Development, Department of Sociology, UNIVERSITAS TRUNOJOYO MADURA, Indonesia; Artificial Intelligence Model for Analyzing the Buying Patterns of Customers, Handbook of Research on AI and Machine Learning Applications in Customer Support and Analytics, IGI Global Scientific Publishing; Restaurant Sales Prediction Using Machine Learning, Handbook of Research on AI and Machine Learning Applications in Customer Support and Analytics, IGI Global Scientific Publishing; S M Nazmuz Sakib’s Hypothesis of Aerosol-Sea Ice Feedback: Implications for Climate System Dynamics, Asian Pacific Journal of Environment and Cancer, Asian Pacific Organization for Cancer Prevention; Assessing the Impact of Arctic Melting in the Predominantly Multilateral World System, Asian Pacific Journal of Environment and Cancer, Asian Pacific Organization for Cancer Prevention; ELECTROCHEMICAL WASTE WATER TREATMENT, Volume 10 and No. 1 (2022) archive, Waste Technology, Waste Resources Research Center (WRRC), Universitas Diponegoro; LiDAR Technology - An Overview, Jan2022, Vol. 15 Issue 1, p36-57, Volume 15 and Issue 1, IUP Journal of Electrical & Electronics Engineering, IUP Publications; The Impact of Oil and Gas Development on the Landscape and Surface in Nigeria, Asian Pacific Journal of Environment and Cancer, Asian Pacific Organization for Cancer Prevention.
Engr. Sakib's online courses include Webflow Animation, Forensic Psychology Basics in Russian, Grundlagen der Soziologie, Philosophy of Astro-Biology, Philosophy of Psychology, and several others published through Eduonix, BitDegree, Teachlr, and Udemy.

References

A Concise Geometry (1921): Durell, Clement v: 9781164352013: Amazon.com: Books. (2010). https://www.amazon.com/Concise-Geometry-Clement-V-Durell/dp/1164352016

Ab, M. (n.d.). Perpendicular bisector theorem. Mathleaks. https://mathleaks.com/study/kb/misc/perpendicular_bisector_theorem

Admin. (2022, September 1). Bisector. BYJUS. https://byjus.com/maths/bisector/#:~:text=If%20a%20point%20is%20equidistant,=%20BF%20and%20AE%20=%20BE.

Advanced Euclidean Geometry. (2007). Google Books. https://books.google.com.bd/books/about/Advanced_Euclidean_Geometry.html?id=4NE6AwAAQBAJ&source=kp_book_description&redir_esc=y

Amazon.com: College Geometry: 9780867204759: Howard Whitley Eves: Books. (n.d.). https://www.amazon.com/College-Geometry-Howard-Whitley-Eves/dp/0867204753

Basic Geometry theorems. (n.d.). former-students.imperial.edu. https://former-students.imperial.edu/055-Bestseller/article?docid=UGP51-4623&title=basic-geometry-theorems.pdf

Blåsjö, V. (2021). Operationalism: an interpretation of the philosophy of ancient Greek geometry. Foundations of Science, 27(2), 587–708. https://doi.org/10.1007/s10699-021-09791-4

Bourne, M. (n.d.). Isosceles triangle theorem. https://www.intmath.com/functions-and-graphs/isosceles-triangle-theorem.php

Calcworkshop. (2020, January 21). Bisector Theorems - Perpendicular vs Angular (2019). https://calcworkshop.com/congruent-triangles/bisector-theorems/

GeeksforGeeks. (2024, March 26). Perpendicular bisector. GeeksforGeeks. https://www.geeksforgeeks.org/maths/perpendicular-bisector/

Khan Academy. (n.d.). https://www.khanacademy.org/math/geometry-home/similarity/angle-bisector-theorem/v/angle-bisector-theorem-proof

Perpendicular Bisector : Learn definition, properties, examples. (n.d.). Testbook. https://testbook.com/maths/perpendicular-bisector#:~:text=A%20perpendicular%20bisector%20of%20a,segment%20they%20bisect%20at%2090.

Puttaswamy, T. (2012a). Mahavira. In Elsevier eBooks (pp. 231–316). https://doi.org/10.1016/b978-0-12-397913-1.00009-0

Puttaswamy, T. (2012b). Sridhara and Prthudakaswami Caturveda. In Elsevier eBooks (pp. 209–229). https://doi.org/10.1016/b978-0-12-397913-1.00008-9

Tall, D., Yevdokimov, O., Koichu, B., Whiteley, W., Kondratieva, M., & Cheng, Y. (2012). Cognitive development of proof. In New ICMI studies series (pp. 13–49). https://doi.org/10.1007/978-94-007-2129-6_2