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@@ Line 1: / Line 1: @@
-[[Image:Morley triangle.svg|right|300px]]
+== We're working on Call of Duty Timberland Schuhe ==
-In [[plane geometry]], '''Morley's trisector theorem''' states that in any [[triangle]], the three points of intersection of the adjacent [[Angle trisection|angle trisectors]] form an [[equilateral triangle]], called the '''first Morley triangle''' or simply the '''Morley triangle'''. The theorem was discovered in 1899 by [[English American|Anglo-American]] [[mathematician]] [[Frank Morley]]. It has various generalizations; in particular, if all of the trisectors are intersected, one obtains four other equilateral triangles.
-==Proofs==
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-There are many [[Mathematical proof|proofs]] of Morley's theorem, some of which are very technical.<ref>{{citation|url=http://www.cut-the-knot.org/triangle/Morley/index.shtml|title=Morley's Miracle|publisher=[[Cut-the-knot]]|last=Bogomolny|first=Alexander|accessdate=2010-01-02}}</ref>
-Several early proofs were  based on delicate [[trigonometry|trigonometric]] calculations.  The first published geometric proof was given by [[M. T. Naraniengar]] in 1909.<ref>{{harvtxt|Coxeter|1967}}.</ref>  Recent proofs include an [[algebra]]ic proof by {{harvs|first=Alain|last=Connes|authorlink=Alain Connes|txt|year=1988|year2=2004}} extending the theorem to general [[Field (mathematics)|fields]], and [[John Horton Conway|John Conway]]'s elementary geometry proof.<ref>[http://www.cut-the-knot.org/triangle/Morley/conway.shtml J. Conway's proof], from Bogomolny.</ref><ref>{{citation|url=http://www.cs.toronto.edu/~mackay/conway.pdf|title=Power|editor1-last=Blackwell|editor1-first=Alan|editor2-last=Mackay|editor2-first=David|editor2-link=David J. C. MacKay|year=2006|chapter=The Power of Mathematics|last=Conway|first=John|author-link=John Horton Conway|publisher=Cambridge University Press|accessdate=2010-10-08|pages=36–50|isbn=978-0-521-82377-7}}</ref>  The latter starts with an equilateral triangle and shows that a triangle may be built around it which will be [[Similarity (geometry)|similar]] to any selected triangle.  Morley's theorem does not hold in [[spherical geometry|spherical]]<ref>[http://lienhard-wimmer.com/applets/dreieck/Morley.html Morley's Theorem in Spherical Geometry], [[Java applet]].</ref> and [[hyperbolic geometry]].
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-[[Image:Morley Proof.svg|thumb|right|480px|Fig 1. &nbsp; Elementary proof of Morley's trisector theorem]]
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-One proof uses the trigonometric identity
-:sin&thinsp;3&theta; &equiv; 4&thinsp;sin&thinsp;&theta;&thinsp;sin(60&deg;+&theta;)&thinsp;sin(120&deg;+&theta;).
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-Points ''D'',''E'',''F'' are constructed on ''BC'' as shown. Clearly α+β+γ = 60° therefore ∠CYA = 120°+β and the angles of Δ''XEF'' are α, 60°+β, 60°+γ. Now sin(60°+β) = ''DX/XE'' and ''AC''/sin(120°+β) = ''AY''/sin&thinsp;γ by the sine rule, so the height ''h'' of Δ''ABC'' is given by
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-:&ensp; = ''AC''&thinsp;sin&thinsp;3&gamma; = 4''AC.AB.DX''&thinsp;sin&thinsp;&gamma;&thinsp;sin&thinsp;&beta; / (''XF.AZ'').
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-As the numerators are equal, ''XE.AY'' = ''XF.AZ''. But ∠EXF = ∠ZAY and the sides about these angles are in the same ratio (because ''XE/XF'' = ''AZ/AY'') so the triangles ''XEF'' and ''AZY'' must be similar. Thus the base angles of Δ''AZY'' are 60°+β and 60°+γ. Similar arguments yield the base angles of Δ''BXZ'' and Δ''CYX'' and all the angles in the figure can now be easily determined.
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-The result can also be proved by the technique of [[reverse reconstruction]].
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-The first Morley triangle has side lengths<ref>Weisstein, Eric W. "First Morley Triangle." From MathWorld--A Wolfram Web Resource. [http://mathworld.wolfram.com/FirstMorleyTriangle.html] </ref>
+== science New Balance 574 ==
-:<math>a^{'}=b^{'}=c^{'}=8R\sin(A/3)\sin(B/3)\sin(C/3), \, </math>
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-where ''R'' is the [[circumradius]] of the original triangle and ''A, B,'' and ''C'' are the angles of the original triangle. Since the area of an equilateral triangle is <math>\tfrac{\sqrt{3}}{4}a'^2,</math> the area can be expressed as
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-:Area <math>= 16 \sqrt{3}R^2\sin^2(A/3)\sin^2(B/3)\sin^2(C/3).</math>
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-==Morley's triangles==
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-Morley's theorem entails 18 equilateral triangles.  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:
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-: ''A''-vertex = 1 : 2 cos(''C''/3) : 2 cos(''B''/3)
-: ''B''-vertex = 2 cos(''C''/3) : 1 : 2 cos(''A''/3)
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-: ''C''-vertex = 2 cos(''B''/3) : 2 cos(''A''/3) : 1
+ </ul>
-Another of Morley's equilateral triangle that is also central triangle is called the '''second Morley triangle''' and is given by these vertices:
-: ''A-''vertex = 1 : 2 cos(''C''/3 &minus; 2π/3) : 2 cos(''B''/3 &minus; 2π/3)
-: ''B-''vertex = 2 cos(''C''/3 &minus; 2π/3) : 1 : 2 cos(''A''/3 &minus; 2π/3)
-: ''C-''vertex = 2 cos(''B''/3 &minus; 2π/3) : 2 cos(''A''/3 &minus; 2π/3) : 1
-The third of Morley's 18 equilateral triangles that is also central triangle is called the '''third Morley triangle''' and is given by these vertices:
-: ''A''-vertex = 1 : 2 cos(''C''/3 &minus; 4π/3) : 2 cos(''B''/3 &minus; 4π/3)
-: ''B''-vertex = 2 cos(''C''/3 &minus; 4π/3) : 1 : 2 cos(''A''/3 &minus; 4π/3)
-: ''C''-vertex = 2 cos(''B''/3 &minus; 4π/3) : 2 cos(''A''/3 &minus; 4π/3) : 1
-The first, second, and third Morley triangles are pairwise [[homothetic]].  Another homothetic triangle is formed by the three points ''X'' on the circumcircle of triangle ''ABC'' at which the line ''XX''<sup>&nbsp;&minus;1</sup> is tangent to the circumcircle, where ''X''<sup>&nbsp;&minus;1</sup> denotes the [[isogonal conjugate]] of ''X''.  This equilateral triangle, called the '''circumtangential triangle''', has these vertices:
-: ''A-''vertex  = csc(''C''/3 &minus; ''B''/3) : csc(''B''/3 + 2''C''/3) : &minus;csc(''C''/3 + 2''B''/3)
-: ''B-''vertex  = &minus;csc(''A''/3 + 2''C''/3) : csc(''A''/3 &minus; C/3) : csc(''C''/3 + 2''A''/3)
-: ''C-''vertex  = csc(''A''/3 + 2''B''/3) : &minus;csc(''B''/3 + 2''A''/3) : csc(''B''/3 &minus; ''A''/3)
-A fifth equilateral triangle, also homothetic to others, is obtained by rotating the circumtangential triangle π/6 about its center.  Called the '''circumnormal triangle''', its vertices are as follows:
-: ''A-''vertex  = sec(''C''/3 &minus; ''B''/3) : &minus;sec(''B''/3 + 2''C''/3) : &minus;sec(''C''/3 + 2''B''/3)
-: ''B-''vertex  = &minus;sec(''A''/3 + 2''C''/3) : sec(''A''/3 &minus; ''C''/3) : &minus;sec(''C''/3 + 2''A''/3)
-: ''C-''vertex  = &minus;sec(''A''/3 + 2''B''/3) : &minus;sec(''B''/3 + 2''A''/3) : sec(''B''/3 &minus; ''A''/3)
-An operation called "extraversion" can be used to obtain one of the 18 Morley triangles from another. Each triangle can be extraverted in three different ways; the 18 Morley triangles and 27 extravert pairs of triangles form the 18 vertices and 27 edges of the [[Pappus graph]].<ref>{{harvtxt|Guy|2007}}.</ref>
-==Related triangle centers==
-The [[centroid]] of the first Morley triangle is given by
-: '''Morley center''' = ''X''(356) = cos(''A''/3) + 2 cos(''B''/3)cos(''C''/3) : cos(''B''/3) + 2 cos(''C''/3)cos(''A''/3) : cos(''C''/3) + 2 cos(''A''/3)cos(''B''/3)
-The first Morley triangle is perspective to triangle ''ABC'', and the perspector is the point
-: '''1st Morley–Taylor–Marr center'''  = ''X''(357) = sec(''A''/3) : sec(''B''/3) : sec(''C''/3)
-==See also==
-*[[Hofstadter points]]
-*[[Morley centers]]
-== Notes ==
-{{reflist|2}}
-== References ==
-*{{citation|first=Alain|last=Connes|authorlink=Alain Connes|url=http://www.numdam.org/item?id=PMIHES_1998__S88__43_0|title=A new proof of Morley's theorem|journal=Publications Mathématiques de l'IHÉS|volume=S88|year=1998|pages=43–46}}.
-*{{citation|first=Alain|last=Connes|authorlink=Alain Connes|url=http://www.ems-ph.org/journals/newsletter/pdf/2004-12-54.pdf|title=Symmetries|journal=[[European Mathematical Society]] Newsletter|volume=54|year=2004|date=December 2004}}.
-*{{citation|first1=H. S. M.|last1=Coxeter|authorlink=Coxeter|first2=S. L.|last2=Greitzer|title=Geometry Revisited|journal=[[The Mathematical Association of America]]|year=1967|lccn=67-20607}}
-*{{citation|first=Richard L.|last=Francis|url=http://www.math-cs.ucmo.edu/~mjms/2002.1/francis9.pdf|title=Modern Mathematical Milestones: Morley's Mystery|journal=Missouri Journal of Mathematical Sciences|volume=14|issue=1|year=2002}}.
-*{{citation
- | last = Guy | first = Richard K. | authorlink = Richard K. Guy
- | mr = 2290364
- | issue = 2
- | journal = [[American Mathematical Monthly]]
- | pages = 97–141
- | title = The lighthouse theorem, Morley & Malfatti—a budget of paradoxes
- | jstor = 27642143
- | url = http://www.math.ucalgary.ca/files/publications/3414848.pdf
- | volume = 114
- | year = 2007}}.
-*{{citation|doi=10.2307/2321680|first1=C. O.|last1=Oakley|first2=J. C.|last2=Baker|title=The Morley trisector theorem|journal=[[American Mathematical Monthly]]|volume=85|issue=9 |year=1978|pages=737–745|jstor=2321680}}.
-*{{citation|first1=F. Glanville|last1=Taylor|first2=W. L.|last2=Marr|title=The six trisectors of each of the angles of a triangle|journal=Proceedings of the Edinburgh Mathematical Society|volume=33|year=1913–14|pages=119–131}}.
-== External links ==
-* [http://mathworld.wolfram.com/MorleysTheorem.html Morleys Theorem] at MathWorld
-* [http://www.mathpages.com/home/kmath376/kmath376.htm Morley's Trisection Theorem] at MathPages
-* [http://demonstrations.wolfram.com/MorleysTheorem/ Morley's Theorem] by Oleksandr Pavlyk, [[The Wolfram Demonstrations Project]].
-[[Category:Triangle geometry]]
-[[Category:Theorems in plane geometry]]

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