On n -sectors of the Angles of an Arbitrary Triangle
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Mathematics in Computer Science
On n-sectors of the Angles of an Arbitrary Triangle Dongming Wang · Bo Huang · Xiaoyu Chen
Received: 21 November 2018 / Revised: 16 November 2019 / Accepted: 27 April 2020 © Springer Nature Switzerland AG 2020
Abstract 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 , 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 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 for n > 3? In this paper, we approach this question using specialized techniques with interactive, semi-automatic algebraic computations and prove that for n = 4 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 , form an equilateral triangle if and only if 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. Keywords Algebraic computation · Angular n-sectors · Equilateral triangle · Morley theorem · Theorem proving Mathematics Subject Classification Primary 68U05; Secondary 68T15
D. Wang · X. Chen (B) SKLSDE-BDBC-LMIB, School of Mathematical Sciences, Beihang University, Beijing 100191, China e-mail: [email protected] D. Wang SMS, School of Mathematics and Physics, Guangxi University for Nationalities, Nanning 530006, China D. Wang Centre National de la Recherche Scientifique, 75794 Paris Cedex 16, France e-mail: [email protected] B. Huang LMIB, School of Mathematical Sciences, Beihang University, Beijing 100191, China e-mail: [email protected]
D. Wang et al.
1 Introduction For an arbitrary triangle ABC, let the two internal trisectors closest to the side AB intersect at point C3 , so do the two closest to the side BC at point A3 and the two closest to the side C A at point B3 . In 1899, Frank Morley discovered that the triangle A3 B3 C3 is always equilateral (cf. Fig. 1), no matter what the triangle ABC looks like [1,2]. Later in 1984, Wu [3,4] provided a machine proof of this elegant theorem, stated in the most general form as follows (see also [5,6]). Morley’s theorem. The neighbouring trisectors of the three angles of an arbitrary triangle intersect to form 27 triangles in all, of which 18 are equilateral. Machine proofs of Morley’s theorem using algebraic methods are often taken as examples to illustrate the surprising success of mechanized geometric the
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