Access the full text.
Sign up today, get DeepDyve free for 14 days.
C. Oakley, J. Baker (1978)
The Morley Trisector TheoremAmerican Mathematical Monthly, 85
Dongming Wang (2001)
Elimination Methods
S-C Chou, Xing Gao, Jason Zhang (1994)
Machine Proofs in Geometry
Wenjun Wu (1986)
Basic principles of mechanical theorem proving in elementary geometriesJournal of Automated Reasoning, 2
Morley Frank (1924)
On the Intersections of the Trisectors of the Angles of a Triangle.
Dongming Wang (2004)
Elimination Practice - Software Tools and Applications
Dongming Wang (1996)
Geometry Machines: From AI to SMC
Bican Xia, L. Yang (2016)
Automated Inequality Proving and Discovering
Hongbo Li (2000)
Vectorial Equations Solving for Mechanical Geometry Theorem ProvingJournal of Automated Reasoning, 25
(2000)
Mathematics Mechanization
S. Chou (1987)
Mechanical Geometry Theorem Proving
Y. Wong, K. Tsang (1982)
A Strong Converse of Morley's Trisector TheoremAmerican Mathematical Monthly, 89
W. Wu (2008)
ON THE DECISION PROBLEM AND THE MECHANIZATION OF THEOREM-PROVING IN ELEMENTARY GEOMETRY
D. Kleven (1978)
Morley's Theorem and a ConverseAmerican Mathematical Monthly, 85
Wen-tsün Wu (1994)
Mechanical Theorem Proving in Geometries: Basic Principles
F. Morley (1900)
On the metric geometry of the plane $n$-lineTransactions of the American Mathematical Society, 1
T. Ida, Asem Kasem, Fadoua Ghourabi, Hidekazu Takahashi (2011)
Morley's theorem revisited: Origami construction and automated proofJ. Symb. Comput., 46
Dongming Wang (1995)
Elimination procedures for mechanical theorem proving in geometryAnnals of Mathematics and Artificial Intelligence, 13
S. Chou, Xiao Gao, Jing-Zhong Zhang (1994)
Machine proofs in geometry - automated production of readable proofs for geometry theorems, 6
Wenjun Wu (1994)
Mechanical Theorem Proving in Geometries
Morley’s theorem shows that the three points, each of which is the intersection of the two internal trisectors that are the closest to the same side of an arbitrary triangle Δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta $$\end{document}, form an equilateral triangle. This beautiful theorem was proved mechanically by Wen-tsün Wu (J. Syst. Sci. Math. Sci. 4:207–235, 1984) in its most general form: the neighbouring trisectors of the three angles of Δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta $$\end{document} intersect to form 27 triangles in all, of which 18 are equilateral triangles, called Morley triangles. A natural question is: does there exist any equilateral triangle, other than Morley triangles, which is formed by three intersection points of the neighbouring angular n-sectors of Δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta $$\end{document} for n>3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n>3$$\end{document}? In this paper, we approach this question using specialized techniques with interactive, semi-automatic algebraic computations and prove that for n=4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n=4$$\end{document} and 5 the three points, each of which is the intersection of the two internal (or two external) angular n-sectors closest to the same side of Δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta $$\end{document}, form an equilateral triangle if and only if Δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta $$\end{document} is equilateral. The computational approach we present can also be applied to other cases for specific n. How to establish the non-existence of equilateral triangles formed by the intersection points of angular n-sectors for general n is a question that remains for further investigation.
Mathematics in Computer Science – Springer Journals
Published: Dec 26, 2020
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.