The idea for this theorem suggested itself to the author from dijkstras proof of morleys theorem 4, as well as a similar result on delaunay triangulations 8. Special and general relativity notes on the michelsonmorley interferometer how it works a monochromatic light wave has one frequency or wavelength. Two singlestarred angle measures set the triangles so that the side oppo. Media in category morley theorem the following 10 files are in this category, out of 10 total. Oct 15, 2014 an interesting theorem of morley s about angle trisectors.
A proof is given of morleys trisector theorem using elementary vector analysis and trigonometry. The triangle described in the trisector theorem above, called the first morley triangle, has vertices given in trilinear coordinates relative to a triangle abc as follows. There are many proofs of morley s trisector theorem 12, 16, 9, 8, 20, 3, 18. Morley s original proof stemmed from his results on algebraic curves tangent to a given number of lines. There are many proofs of morley s theorem, some of which are very technical. Sep 27, 2017 a new approach is shown that mechanically proves various theorems in plane geometry by recasting them in terms of constraint satisfaction.
In this paper we are taking the rst steps towards extending morley s categoricity theorem \to. Weshall nowshow thatingeneral therotation gwhich permutes cyclically the points. Trigg 281 an elementary geometric proof of the morley theorem dan sokolowsky 291 the beauty and truth of the morley theorem leon bankoff 294. There are many different proofs of morley s theorem. Morleys triangle and morleys man jim wilsons home page. Theorem list alphabetical this version of the complete list of theorems is given alphabetically by keyword. Notes on morleys proof of his theorem on angle trisectors. One has g3 1g 3 2g 3 3 1 since each g3 i can be expressed as the product of the symmetries.
His proof is considered the simplest and a very good example of a backward proof. As usual in mathematics, numerous attempts have been made to find a simple, elementary proof that could match the level of knowledge and proficiency required to grasp the statement of the theorem. Morleys theorem states that for any triangle, the intersections of its adjacent angle trisectors form an equilateral triangle. This key insight lets us piece together the following argument, where we build up the diagram backwards from its constituent pieces. A simple geometric proof of morleys trisector theorem bloggen. In other words, the lines am, bl, and ck are concurrent. Special and general relativity notes on the michelsonmorley. Theorems with names in geometry oulun lyseon lukio. Thus albertbrauerhassenoether main theorem will appear under a for albert, b for brauer, h for hasse, n for noether and m for main but not t for theorem. Science day celebration morleys miracle frank morley usa discovered a theorem about triangle in 1899, approximately 2000 years after rst theorems about triangle were published by euclid. Mar 18, 2012 well, morleys theorem tells us that this diagram has three nice 60degree angles in the middle, but we may suspect that, in fact, all of the angles are nice. Pdf among the dozens of proofs of morleys theorem, the proofs of j. All files are pdf, mostly between 100 and 300 kbytes in size.
Morleys categoricity theorem states that if a theory of rst order logic is categorical in some uncountable power, then it is categorical in every uncountable power. Matematicas visuales john conways proof of morleys theorem. Trisectors like bisectors with equilaterals instead of points scielo. The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle. Finally, it is worth mentioning that morleys categoricity theorem may be of historical.
Take k c and let g 1 be the rotation with center a and angle 2a,where3a is the angle bac and similarly for g 2, g 3. Generalizing morleys and other theorems with automated. Theorem of the day is registered as a uk trademark, no. Let the angle bac be trisected by lines ab and ac, in that order. Jul 10, 2009 morley s theorem asserts that the points of intersection of the adjacent angle trisectors of the angles of an arbitrary triangle abc are the vertices of an equilateral triangle def. The construction of morleys triangle by the straightedge and compass method is impossible because of the wellknown impossibility result for angle trisection. The three points of intersection of adjacent trisectors of the angles of any triangle form an equliateral triangle.
Morleys theorem is one of the most surprising and attractive twentieth century results in plane geometry. Its simplicity is part of its beauty, but could easily lead us. Pdf is the mystery of morleys trisector theorem resolved. The intersection points of the trisectors of the angles of a triangle intersect each other in the vertices of an equilateral triangle.
Theorem of the day morleys miracle let a, b, c be the vertices of a triangle. Morley s trisector theorem states that the points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle 10. Morleys categoricity theorem department of mathematics. The morley trisector theorem ufficio digital learning e multimedia. A theory t is called categorical if any two models of t of cardinality are isomorphic. Morley rank and strongly minimal structures are key tools in the proof of morleys categoricity theorem and in the larger area of stability theory. This was a surprising discovery made by frank morley 1899. Pdf is john conways proof of morleys theorem the simplest. In this paper we are taking the rst steps towards extending morleys categoricity theorem \to the nite. The categoricity theorem 20 acknowledgments 20 references 20 a theory is called categorical, or categorical in power, if it has one model up to isomorphism of cardinality. Under the hypothesis of the theorem, we can assume that the. Recent proofs include an algebraic proof by alain connes 1998, 2004 extending the theorem to general fields other than characteristic three, and john conway s elementary geometry proof. Morley s theorem states that a theory which is categorical for an uncountable cardinal is categorical in all uncountable cardinals. Morleys theorem does not hold in noneuclidean geometry, so that the proof should make.
Morleys mystery pdf, missouri journal of mathematical sciences, 14 1. An expanded version of this book is available under the title model theory. Thus, and whatever the characteristic of k is, we get that a. I hope that morley s trisection theorem has many applications. A python 3 implementation called geopar affords transparent proofs of wellknown theorems as well as new ones, including a generalization of morleys theorem. My problem with the theorem is that i havent found a significant example in which this theorem can be applied, in which no other argument has been found. The known expression for the side of morleys equilateral triangle is also obtained. As the above table suggests, we have three types of triangle.
In fact, the centroid of the morley s triangle is known as the first morley s center see figure2. Let as be the set of angles in the triangles of an stpf s i. A russian copy of poizats book may be downloaded and you can write en francais to the author to buy a copy of the book in french. A shorter model theory by wilfrid hodges, cambridge university press, 1997. An interesting property happens when we connect the opposite vertices of the original triangle and its morley s triangle.
In plane geometry, morleys trisector theorem states that in any triangle, the three points of. Morleys proof canadian mathematical society mitacs winnipeg. Connes to explore possible generalizations of morley s trisector theorem to triangles in arbitrary value fields. Morley proved this as a corollary to some other very complicated calculations of conic sections i think based on certain properties of triangles. Several early proofs were based on delicate trigonometric calculations. Well, morleys theorem tells us that this diagram has three nice 60degree angles in the middle, but we may suspect that, in fact, all of the angles are nice. The equilateral triangle just set it to have sidelength 1. Tran quang hung a direct trigonometric proof of morleys. In particular, we prove morleys theorem is, in fact, a natural consequence of an overlooked simple result concerning a general property of angle bisectors as loci.
A new proof of morleys theorem pdf, alain connes juan fernando. Morleys miracle morleys theorem states that the intersections of the trisectors of the angles of a triangle form the vertices of an equilateral triangle. Clicking on a theorem will open its description page pdf. This theorem has many, many proofs and extensions 8 has over 100 references, which would make ne material for a future \what is. Tran quang hung a direct trigonometric proof of morleys theorem. Morley 273 notes on morley s proof of his theorem on angle trisectors dan pedoe 276 robson s proof of morley s theorem 280 a list of references to the morley theorem co. Though we dont give the actual proof, which involves some long factoring, we will be able to observe how connes theorem is applicable to solve the given problem. Angle trisection, morleys theorem, morley trisector theorem. Morley s theorem is renowned as being a theorem that s really hard to prove. Morleys proof canadian mathematical society mitacs winnipeg june 3, 2007 john t. In this paper we discuss with some reasons why the above possible.
397 1278 256 541 412 647 208 1102 64 1383 868 300 1130 421 996 1298 980 960 718 311 69 809 599 178 1510 158 313 1147 267 109 1082 87 1413 1187