Talk:Morley's trisector theorem

message: Historical correction: Where is the justification for the sentence "Morley originally did not prove the result leaving it as a conjecture"? This is not the case. 
Frank Morley realized that the simple statement usually known as Morley's Theorem follows from much more general considerations involving cardioids that were of interest to him, and the subject of his papers beginning with [On the metric properties of the plane $n$-line, Trans. Amer. Math. Soc. 1(1900)97--115; 8(1907),14--24, http://www.ams.org/tran/].  Later he included the simpler elementary form of the theorem in his book written with his son, also Frank, Inversive Geomettry from 1933.  
Henri Lebesgue in his exegesis of these matters [Sur les n-sectrices d'un triangle, Ens. Math, 38 (1939--1940), 39--58, http://retro.seals.ch/cntmng?type=pdf&rid=ensmat-001:1939-1940:38::13&subp=hires] remarks that Morley obtained the theorem 'incidentally' to other research, but he is offering years later another sort of proof just so the result may be well understood.  
Glanville Taylor and Marr, in the paper referenced on the main page here, The six trisectors of each of the angles of a triangle  
(<doi:10.1017/S0013091500035100> http://journals.cambridge.org/action/displayIssue?iid=3162136#), say they only had heard of the special form of Morley's results for triangles and trisectors, and so they provide two geometrical proofs (one due to W. E. Philip) and a generalization; another paper of each of the authors on the subject follows immediately in the same journal (<doi:10.1017/S0013091500035112> and <doi:10.1017/S0013091500035124>).  Nonetheless, they are careful to state at the start that Morley's result, proved 14 years before, contains the special case of interest, though they add further illumination to the situation.  
Indeed their style of labelling the points and triangles seems to have been followed by Lebesgue, and then by M.-B. Gambier [Trisectrices des angles d'un triangle, Ann. Sci de l' É. N. S. 3e sér., 71. no 2 (1954), 191--212 <http://www.numdam.org/item?id=ASENS_1954_3_71_2_191_0>], who, writing about 50 years after the fact, does suggest Morely "hardly sketched the special case" which "he'd demonstrated in his lectures at Johns Hopkins University".  Gambier's notation and diagrams, and the notion of rays sweeping out areas ("...les trois secteurs balayés par les demi-droites...", p. 193 l. -5) seem to me to prefigure Richard Guy's very nice presentation (influenced by collaboration with John H. Conway) in terms of lighthouse geometry (as referenced on the main page, although the link there is to a preprint version missing page 4 and rather shorter than the finally published [Amer. Math. Monthly  114  (2007),  no. 2, 97--141; MR2290364] which has almost twice as many references).  
I have not seen some of the earlier French papers referred to by Lebesgue, nor Morley's 1924 paper in a Japanese journal, where the simple form of the theorem may be plainly stated.  So there may be more information to be had about the matter.  I will try to remove the sentence mentioned at the start if nobody offers any good explanation as to why not.
David P Fraser (talk) 15:38, 10 March 2010 (UTC)
thread_title:  Historical correction 

message: Seems pretty opinionated to me: most surprising and beautiful, arguably the best proof etc. --24.85.75.88 (talk) 04:48, 9 September 2011 (UTC)
thread_title:  Opinionated?  

message: I hope that Morley's trisection theorem has many applications. Ashachaaru (talk) 18:16, 13 November 2017 (UTC)
thread_title:  Applications of Morley's trisection theorem 

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