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</div> <div id="siteSub" class="noprint">Na Galipedia, a Wikipedia en galego.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="gl" dir="ltr"><p>Os <b>axiomas de Birkhoff</b> son un conxunto de catro <a href="/wiki/Axioma" title="Axioma">postulados</a> establecidos por <a href="/wiki/George_David_Birkhoff" title="George David Birkhoff">G. D. Birkhoff</a> en 1932 para a <a href="/wiki/Xeometr%C3%ADa_euclidiana" title="Xeometría euclidiana">xeometría euclidiana</a> no plano.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>1<span>]</span></a></sup> Estes postulados están todos baseados na <a href="/wiki/Xeometr%C3%ADa" title="Xeometría">xeometría</a> básica que pode ser confirmada experimentalmente cunha <a href="/w/index.php?title=Vernier_Escala&action=edit&redlink=1" class="new" title="Vernier Escala (a páxina aínda non existe)">escala</a> e un <a href="/w/index.php?title=Protractor&action=edit&redlink=1" class="new" title="Protractor (a páxina aínda non existe)">transportador</a>. Como os postulados se constrúen sobre os <a href="/wiki/N%C3%BAmero_real" title="Número real">números reais</a>, o enfoque é semellante a unha introdución baseada nun modelo da xeometría euclidiana. </p><p>O sistema de axiomas de Birkhoff foi utilizado no libro de texto da escola secundaria de Birkhoff e Beatley.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>2<span>]</span></a></sup> Estes axiomas tamén foron modificados polo School Mathematics Study Group para proporcionar un novo estándar para o ensino da xeometría nun instituto de secundaria, coñecidos como axiomas SMSG.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>3<span>]</span></a></sup> </p><p>Algúns outros libros de texto dos fundamentos da xeometría usan variantes dos axiomas de Birkhoff .<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span>[</span>4<span>]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Postulados">Postulados</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiomas_de_Birkhoff&veaction=edit&section=1" title="Editar a sección: «Postulados»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Axiomas_de_Birkhoff&action=edit&section=1" title="Editar o código fonte da sección: Postulados"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A distancia entre dous puntos <span class="texhtml"><i>A</i></span> e  <span class="texhtml">B</span> é denotada por  <span class="texhtml"><i>d</i>(<i>A, B</i>)</span>, e o ángulo formado por tres puntos <span class="texhtml"><i>A</i>, <i>B</i>, <i>C</i></span> denótase como <span class="texhtml">∠ <i>ABC</i></span>. </p> <ul><li><b>Postulado I: Postulado de medida do segmento</b>: O conxunto de puntos<span class="texhtml">{<i>A</i>, <i>B</i>, ...</span>} sobre unha liña pódese poñer nunha correspondencia 1:1 cos <a href="/wiki/N%C3%BAmero_real" title="Número real">números reais</a>  De modo que <span class="texhtml">|<i>b</i> − <i>a</i>| = <i>d</i>(<i>A, B</i>)</span>  para todos os puntos <span class="texhtml"><i>A</i></span> e <span class="texhtml"><i>B</i></span>.</li> <li><b>Postulado II : Postulado da recta por dous puntos</b>: Hai unha e só unha liña <span class="texhtml"><i>ℓ</i></span> que contén calquera dous puntos distintos dados <span class="texhtml"><i>P</i></span> e <span class="texhtml"><i>Q</i></span>.</li> <li><b>Postulado III : Postulado de medida do ángulo</b>: O conxunto de semirectas <span class="texhtml">{<i>ℓ, m, n</i>, ...</span>} que parten dun punto calquera <span class="texhtml"><i>O</i></span> pode poñerse en correspondencia 1:1 cos números reais <span class="texhtml"><i>a</i> (mod 2<i>π</i>)</span> así que se <span class="texhtml"><i>A</i></span> e <span class="texhtml"><i>B</i></span> son puntos de <span class="texhtml"><i>ℓ</i></span> e <span class="texhtml"><i>m</i></span> respectivamente (distintos de <span class="texhtml"><i>O</i></span>), a diferenza <span class="texhtml"><i>a</i><sub><i>m</i></sub> − <i>a</i><sub><i>ℓ</i></sub> (mod 2π)</span> dos números asociados coas liñas <span class="texhtml"><i>ℓ</i></span> e <span class="texhtml"><i>m</i></span>, é <span class="texhtml">∠ <i>AOB</i></span>. Ademais, se o punto <span class="texhtml"><i>B</i></span> en <span class="texhtml"><i>m</i></span> varía continuamente nunha liña <span class="texhtml"><i>r</i></span> que non contén o vértice <span class="texhtml"><i>O</i></span>, o número <span class="texhtml"><i>a</i><sub><i>m'</i></sub></span><i> tamén varía continuamente.</i></li> <li><b>Postulado IV: Postulado de semellanza</b>: Dados dous triángulos <span class="texhtml"><i>ABC</i></span> e <span class="texhtml"><i>A'B'C' </i></span> e unha constante <span class="texhtml"><i>k</i> > 0</span> tal que <span class="texhtml"><i>d</i>(<i>A', B' </i>) = <i>kd</i>(<i>A, B</i>), <i>d</i>(<i>A', C' </i>) = <i>kd</i>(<i>A, C</i>)</span> e <span class="texhtml">∠ <i>B'A'C' </i> = ±∠ <i>BAC</i></span>  entón <span class="texhtml"><i>d</i>(<i>B', C' </i>) = <i>kd</i>(<i>B, C</i>), ∠ <i>C'B'A' </i> = ±∠ <i>CBA</i></span>  e   <span class="texhtml">∠ <i>A'C'B' </i> = ±∠ <i>ACB</i></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notas">Notas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiomas_de_Birkhoff&veaction=edit&section=2" title="Editar a sección: «Notas»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Axiomas_de_Birkhoff&action=edit&section=2" title="Editar o código fonte da sección: Notas"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><cite class="citation book">Birkhoff, George David. <i>A Set of Postulates for Plane Geometry (Based on Scale and Protractors)</i>. pp. 329–345.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3AAxiomas+de+Birkhoff&rft.aufirst=George+David&rft.aulast=Birkhoff&rft.btitle=A+Set+of+Postulates+for+Plane+Geometry+%28Based+on+Scale+and+Protractors%29&rft.genre=book&rft.pages=329-345&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/George_David_Birkhoff" title="George David Birkhoff">Birkhoff, George David</a>; Beatley, Ralph (2000) [first edition, 1940]. <i>Basic Geometry</i> (3rd ed.). American Mathematical Society. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:Fontes_bibliogr%C3%A1ficas/978-0-8218-2101-5" title="Especial:Fontes bibliográficas/978-0-8218-2101-5">978-0-8218-2101-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3AAxiomas+de+Birkhoff&rft.au=Beatley%2C+Ralph&rft.aufirst=George+David&rft.aulast=Birkhoff&rft.btitle=Basic+Geometry&rft.date=2000&rft.edition=3rd&rft.genre=book&rft.isbn=978-0-8218-2101-5&rft.pub=American+Mathematical+Society&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130615062200/http://userpages.umbc.edu/~rcampbel/Math306/Axioms/SMSG.html">"SMSG axioms"</a>. Arquivado dende <a rel="nofollow" class="external text" href="http://userpages.umbc.edu/~rcampbel/Math306/Axioms/SMSG.html">o orixinal</a> o 15 de xuño de 2013<span class="reference-accessdate">. Consultado o 15 de xuño de 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3AAxiomas+de+Birkhoff&rft.btitle=SMSG+axioms&rft.genre=unknown&rft_id=http%3A%2F%2Fuserpages.umbc.edu%2F~rcampbel%2FMath306%2FAxioms%2FSMSG.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><cite class="citation book">Kelly, Paul Joseph; Matthews, Gordon (1981). <a rel="nofollow" class="external text" href="https://archive.org/details/noneuclideanhype0000kell"><i>The non-Euclidean, hyperbolic plane: its structure and consistency</i></a>. Springer-Verlag. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:Fontes_bibliogr%C3%A1ficas/0-387-90552-9" title="Especial:Fontes bibliográficas/0-387-90552-9">0-387-90552-9</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3AAxiomas+de+Birkhoff&rft.au=Matthews%2C+Gordon&rft.aufirst=Paul+Joseph&rft.aulast=Kelly&rft.btitle=The+non-Euclidean%2C+hyperbolic+plane%3A+its+structure+and+consistency&rft.date=1981&rft.genre=book&rft.isbn=0-387-90552-9&rft.pub=Springer-Verlag&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnoneuclideanhype0000kell&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Véxase_tamén"><span id="V.C3.A9xase_tam.C3.A9n"></span>Véxase tamén</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiomas_de_Birkhoff&veaction=edit&section=3" title="Editar a sección: «Véxase tamén»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Axiomas_de_Birkhoff&action=edit&section=3" title="Editar o código fonte da sección: Véxase tamén"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Outros_artigos">Outros artigos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Axiomas_de_Birkhoff&veaction=edit&section=4" title="Editar a sección: «Outros artigos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Axiomas_de_Birkhoff&action=edit&section=4" title="Editar o código fonte da sección: Outros artigos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Xeometr%C3%ADa_euclidiana" title="Xeometría euclidiana">Xeometría euclidiana</a></li> <li><a href="/wiki/Espazo_euclidiano" title="Espazo euclidiano">Espazo euclidiano</a></li> <li><a href="/w/index.php?title=Fundamentos_da_xeometr%C3%ADa&action=edit&redlink=1" class="new" title="Fundamentos da xeometría (a páxina aínda non existe)">Fundamentos da xeometría</a></li> <li><a href="/wiki/Axiomas_de_Hilbert" title="Axiomas de Hilbert">Axiomas de Hilbert</a></li> <li><a href="/wiki/Axiomas_de_Tarski" title="Axiomas de Tarski">Axiomas de Tarski</a></li></ul> <div role="navigation" class="navbox" aria-labelledby="Control_de_autoridades" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="Control_de_autoridades" scope="row" class="navbox-group" style="width:1%;width: 12%; 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