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TitleMorley’s theorem revisited: Origami construction and automated proof
Author(s) Fadoua Ghourabi, Asem Kasem, Hidekazu Takahashi, Thomas Uchida
TypeArticle in Journal
AbstractMorley’s theorem states that for any triangle, the intersections of its adjacent angle trisectors form an equilateral triangle. The construction of Morley’s triangle by the straightedge and compass method is impossible because of the well-known impossibility result for angle trisection. However, by origami, the construction of an angle trisector is possible, and hence that of Morley’s triangle. In this paper we present a computational origami construction of Morley’s triangle and an automated correctness proof of the generalized Morley’s theorem. During the computational origami construction, geometrical constraints in symbolic representation are generated and accumulated. Those constraints are then transformed into algebraic forms, i.e. a set of polynomials, which in turn are used to prove the correctness of the construction. The automated proof is based on the Gröbner bases method. The timings of the experiments of the Gröbner bases computations for our proofs are given. They vary greatly depending on the origami construction methods, the algorithms for the Gröbner bases computation, and variable orderings.
KeywordsMorley’s theorem, Computational origami, Automated geometrical theorem proving, Gröbner bases
URL http://www.sciencedirect.com/science/article/pii/S0747717110001768
JournalJournal of Symbolic Computation
Pages571 - 583
NoteGroebner Bases and Applications
Translation No
Refereed No