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What to do when the trisector comes

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... Considering either or as the acute angle to be trisected, it is required that all the points defining the "trisection angles" be defined in a two dimensional system (plane geometry). Thus the problem of trisecting an angle in general has to be sought following the classical rule s of Euclidean geometry, and not using the mechanical methods being employed [10]. ...
... 3. Using circles of radius and the ray , trisect the chord at points and as shown in figure (10). See trisection example on annex-1. ...
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This paper is devoted to exposition of a provable classical solution for the ancient Greek’s classical geometric problem of angle trisection [3]. (Pierre Laurent Wantzel, 1837),presented an algebraic proof based on ideas from Galois field showing that, the angle trisection solution correspond to an implicit solution of the cubic equation; , which he stated as geometrically irreducible [23]. The primary objective of this novel work is to show the possibility to solve the trisection of an arbitrary angle using the traditional Greek’s tools of geometry, and refutethe presented proof of angle trisection impossibility statement. The exposedproof of the solution is theorem , which is based on the classical rules of Euclidean geometry, contrary to the Archimedes proposition of usinga marked straightedge construction [4], [11].
... Through the year 2017, several researchers; [5], and [6], have published a high profile scientific refutes against the angle trisection impossibility statement. [9] and [14] provides a clear description of Euclidean geometry, interpretation of the governing rules, and the differences between Euclidean geometry and Solid geometry, together with the other types of geometry, wellconstructed. A clear account of Euclidean and non-Euclidean geometry is found in the treatise; the Euclid's elements, translated and interpreted by different mathematics generations into various fashions as in [11]. ...
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This paper presents an elegant classical geometric solution to the ancient Greek's problem of angle trisection. Its primary objective is to provide a provable construction for resolving the trisection of an arbitrary angle, based on the restrictions governing the problem. The angle trisection problem is believed to be unsolvable for compass-straightedge construction. As stated by Pierre Laurent Wantzel (1837), the solution of the angle trisection problem corresponds to an implicit solution of the cubic equation x cubed minus 3x minus 1 equals 0, which is algebraically irreducible, and so is the geometric solution of the angle trisection problem. The goal of the presented solution is to show the possibility to solve the trisection of an arbitrary angle using the traditional Greek's tools of geometry (a classical compass and straightedge) by changing the problem from the algebraic impossibility classification to a solvable plane geometrical problem. Fundamentally, this novel work is based on the fact that algebraic irrationality is not a geometrical impossibility. The exposed methods of proof have been reduced to the Euclidean postulates of classical geometry.
... Considering the trisection of and angle as a cubic equation translated the problem from a 2 (plane geometry) problem as it should be sought, to a 3 (solid geometry) problem, which involve equations of the form 3 as discussed in section 1.2 . This was a serious misconstruction and due to inability to geometrically solve the cubic equations, and the fact that no geometrical algorithm has been presented to solve the partitioning of an angle into a given ratio, mathematicians wicked to pseudo mathematical approaches which do not redress the problem with the desired degree of correctness [9]. This paper relies on a simple concept of the Archimedes theorem of straightedge which stated "If we are in possession of a straightedge that is notched in two places, then it is possible to trisect an arbitrary angle", by revealing a geometrical solution for this ancient problem, contrally to the Archimedes approach of using a marked straightedge. ...
... The unexpectedly simple solution shown in Fig.3a fully justifies the intuition of all Trisectors [14,15] amongst both professional and amateur mathematicians since Wantzel's paper of 1837 who believed that the solution of the trisection problem may indeed be possible. ...
Conference Paper
A solution of the ancient Greek problem of trisection of an arbitrary angle employing only compass and straightedge and its algebraic proof are presented. It is shown that while Wantzel's theory of 1837 concerning irreducibility of the cubic x3-3x-1=0 is correct it does not imply impossibility of trisection of arbitrary angle since rather than a cubic equation the trisection problem is shown to depend on the quadratic equation y2-3+c=0 where c is a constant. The earlier formulation of the problem by Descartes the father of algebraic geometry is also discussed. If one assumes that the ruler and the compass employed in the geometric constructions are in fact Platonic ideal instruments then the trisection solution proposed herein should be exact.
... The unexpectedly simple solution shown inFig.3a fully justifies the intuition of all Trisectors [14, 15] amongst both professional and amateur mathematicians since Wantzel's paper of 1837 who believed that the solution of the trisection problem may indeed be possible. ...
Article
A solution of the ancient Greek problem of trisection of an arbitrary angle employing only compass and straightedge and its algebraic proof are presented. It is shown that while Wantzel's [1] theory of 1837 concerning irreducibility of the cubic x3 - 3x - 1 = 0 is correct it does not imply impossibility of trisection of arbitrary angle since rather than a cubic equation the trisection problem is shown to depend on the quadratic equation y2 - 3 + c = 0 where c is a constant. The earlier formulation of the problem by Descartes the father of algebraic geometry is also discussed. If one assumes that the ruler and the compass employed in the geometric constructions are in fact Platonic ideal instruments then the trisection solution proposed herein should be exact.
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