Talk:Morley's trisector theorem

Where is the proof
Where's the proof? -- Walt e r Humala God   save him! (wanna Talk?) 17:12, 18 January 2007 (UTC)
 * I just added references that link to proofs. Mathman1550 (talk) 21:04, 6 May 2008 (UTC)
 * Dear Mathman,
 * I don't know either way, but am wondering about the assertion that the theorem does not work on the sphere (& maybe a general hyperbolic surface. Since the problem (not planar solutions) does not seem to involve parallels (i.e. Postulate V), it seems to me that it should project over to the sphere.  (I lack the background to make a similar guess about hyperbolic geometry.)
 * When I first saw Conway's marvelous proof, I wondered about the spherical counterpart. Seeking a quick rejection on the sphere, I ran several examples using a calculator.  I was not able to find one that did not work _within_ the accuracy of the calculator; so no decvisive result.
 * I did not attempt a general solution using spherical trigonometry. (I think it looked like it was going to get nasty due to the trisected angles.)
 * So, I guess my question is, "Who disproved it for the sphere, and is that work available." {One possibility is that the planar angle trisectors do not result in spherical angle trisectors, and this difference might have been too slight for the limited number of examples that I tried.
 * Thank you for any clarification.
 * Best wishes,
 * Lem Chastain (talk) Ljc 17:06, 9 May 2023 (UTC)
 * Dear Mathman, For 5/9/2023 reply, I had forgotten how to alert the original writer. I hope this one is right.  If so, please see my prior question. mathman1550 (talk)
 * Thank you, Lem(talk) Ljc 14:57, 12 May 2023 (UTC)
 * Darn; trying agian. Please see question two replies back.  Thank you, Lem   u|matnman1550  (talk) Ljc 15:09, 12 May 2023 (UTC)
 * You are responding to a post that is more than a decade old, made by a user who hasn't been active on Wikipedia in many years. You aren't going to get a response. MrOllie (talk) 15:33, 12 May 2023 (UTC)
 * Thanks for that! At least you replied; so your last sentence is a bit of a contradiction. However, I get your point.  Know any Lobachevskians?  [Good exercise for recalling "ping" format though.]  u|MrOllie (talk) Ljc 20:13, 12 May 2023 (UTC)
 * Or maybe within, " "?
 * Lem Chastain (talk) Ljc 20:15, 12 May 2023 (UTC)
 * For what happens in hyperbolic geometry, see
 * They give some messy formulas for the side lengths of the Morley triangle and observe that they are not symmetric. —David Eppstein (talk) 22:16, 12 May 2023 (UTC)
 * Thank you Mr. Eppstein. It is not what I was hoping for.   I should have stuck with the spherical case:  the one in which I had tried a few examples on a calculator, with inconclusive results. (Disproving it for the sphere with a calculator might be possible, but proving it -- or a computer -- probably impossible.  Exact calculations are also messy on the sphere, but probably not as bad as in the hyperbolic case.   (talk) Ljc 03:13, 13 May 2023 (UTC)
 * BTW, if you just want to call me by name, that's fine, but if you want to use a title, in this sort of topic, "Mr." is the wrong one to use. —David Eppstein (talk) 04:23, 13 May 2023 (UTC)
 * Dear David, Please pardon my presumption. -- Lem (talk) Ljc 17:03, 13 May 2023 (UTC)

Historical correction
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 ( 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 ( and ). 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 ], 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)

Opinionated?
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)

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

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