Sylvester–Gallai theorem (original) (raw)
El teorema de Sylvester-Gallai estableix que donat un conjunt finit de punts no alineats en el pla euclidià, existeix una recta que conté exactament dos dels punts. Aquest enunciat el va proposar el matemàtic anglès James Joseph Sylvester el 1893. Ho va fer en forma de problema a una columna de la revista . Uns quaranta anys després, Tibor Gallai, un matemàtic hongarès, va demostrar el teorema.
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| dbo:abstract | El teorema de Sylvester-Gallai estableix que donat un conjunt finit de punts no alineats en el pla euclidià, existeix una recta que conté exactament dos dels punts. Aquest enunciat el va proposar el matemàtic anglès James Joseph Sylvester el 1893. Ho va fer en forma de problema a una columna de la revista . Uns quaranta anys després, Tibor Gallai, un matemàtic hongarès, va demostrar el teorema. (ca) Der Satz von Sylvester-Gallai ist ein mathematischer Satz der ebenen Geometrie. Er besagt, dass zu jeder endlichen Menge von Punkten, die nicht alle auf einer Geraden, aber in einer Ebene liegen, eine Gerade existiert, die genau zwei Punkte der Menge enthält. Benannt ist er nach James Joseph Sylvester, der die Aussage 1893 in der Educational Times erstmals formulierte, und Tibor Gallai, der 1944 den ersten Beweis veröffentlichte, nachdem Paul Erdős das Problem 1943 neu stellte. Der erste bekannte Beweis stammt von Eberhard Melchior aus dem Jahre 1940. (de) في الهندسة الرياضية، تنص مبرهنة سلفستر غالاي (بالإنجليزية: Sylvester–Gallai theorem) على أنه في حال وجد مجموعة منتهية من النقاط في المستوي فإن هذه النقاط إما أن: 1. * تكون جميع النقاط واقعة على مستقيم واحد 2. * يوجد مستقيم يحتوي تماماً اثنتين من هذه النقاط. تم تقديم هذه المبرهنة من قبل جايمس جوزيف سلفستر في عام 1893، وبشكل منفصل من قبل بول إيردوس في عام 1943 ومن ثم برهنت من قبل في عام 1944. هذه المبرهنة لا تنطبق على مجموعة غير منتهية من النقاط. (ar) En géométrie discrète, le théorème de Sylvester-Gallai affirme qu'étant donné un ensemble fini de points du plan, on a l'alternative suivante : * soit tous les points sont alignés, * soit il existe une droite qui contient exactement deux de ces points. Les droites contenant exactement deux points sont nommées droites ordinaires. Ce théorème ne s'applique pas à des ensembles infinis de points : il suffit pour s'en convaincre de considérer l'ensemble des points de coordonnées entières dans le plan euclidien. (fr) The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944. A line that contains exactly two of a set of points is known as an ordinary line. Another way of stating the theorem is that every finite set of points that is not collinear has an ordinary line. According to a strengthening of the theorem, every finite point set (not all on one line) has at least a linear number of ordinary lines. An algorithm can find an ordinary line in a set of points in time . (en) Il teorema di Sylvester–Gallai (in origine una congettura nota come problema di Sylvester) afferma che, dato un insieme finito di almeno 3 punti del piano, non è possibile disporli in una configurazione tale che ogni retta che passi per due punti ne contenga anche un terzo, fatto salvo il caso in cui siano tutti allineati. In altri termini, è vera la seguente alternativa: 1. * o tutti i punti sono allineati; 2. * oppure esiste almeno una retta che contiene solo due punti dell'insieme. Questo enunciato, molto intuitivo e di semplice formulazione, fu proposto come problema aperto da James Joseph Sylvester nel 1893 e risolto solo nel 1944 da Tibor Gallai. Una versione più quantitativa dell'enunciato è il . L'enunciato non è vero per un insieme di infiniti punti del piano: un controesempio piuttosto evidente è fornito dall'insieme . (it) O teorema de Sylvester–Gallai afirma que dado um número finito de pontos no plano euclidiano, das duas uma: 1. * todos os pontos são colineares (estão em linha reta); ou 2. * há uma reta que passa por exatamente dois dos pontos. Esta afirmação foi conjecturada por James Joseph Sylvester, e foi o primeiro a prová-la. Sem saber da prova de Melchior, Paul Erdős a enunciou como conjectura, que então foi provada por Tibor Gallai, e logo em seguida por outros. Uma demonstração encontrada por em 1948 foi bastante apreciada por Erdős. (pt) Теорема Сильвестра — классический результат комбинаторной геометрии о конфигурациях прямых на плоскости. (ru) Sylvesters sats är en matematisk sats uppkallad efter matematikern James Joseph Sylvester som lyder Om man har en ändlig punktmängd i planet, , där alla punkter inte ligger längs en linje så finns det en linje som skär exakt två punkter. (sv) Теорема Сильвестра — класичний результат комбінаторної геометрії про конфігурації прямих на площині. (uk) 西爾維斯特–高洛伊定理（Sylvester–Gallai theorem）說明若在平面上有有限數目的點，點的數目多於2，它們不是全部共線，有一條線上剛好有兩點，如果过任意两点的直线都必过第三点，则所有的点共线。 這個定理在無限點的情況並不成立，可以考慮格點。 (zh) |
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| dbp:author1Link | Nicolaas Govert de Bruijn (en) |
| dbp:author2Link | Paul Erdős (en) |
| dbp:authorLink | H. J. Woodall (en) |
| dbp:authorlink | Paul Erdős (en) Harold Scott MacDonald Coxeter (en) James Joseph Sylvester (en) Gabriel Andrew Dirac (en) Eberhard Melchior (en) Theodore Motzkin (en) |
| dbp:first | Gabriel (en) Paul (en) Theodore (en) J. J. (en) Eberhard (en) H. J. (en) H. S. M. (en) |
| dbp:last | Woodall (en) Sylvester (en) Melchior (en) de Bruijn (en) Erdős (en) Coxeter (en) Dirac (en) Motzkin (en) |
| dbp:mode | cs2 (en) |
| dbp:title | Ordinary Line (en) |
| dbp:urlname | OrdinaryLine (en) |
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| dbp:year | 1893 (xsd:integer) 1941 (xsd:integer) 1943 (xsd:integer) 1948 (xsd:integer) 1951 (xsd:integer) 1969 (xsd:integer) |
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| gold:hypernym | dbr:Collinear |
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| rdfs:comment | El teorema de Sylvester-Gallai estableix que donat un conjunt finit de punts no alineats en el pla euclidià, existeix una recta que conté exactament dos dels punts. Aquest enunciat el va proposar el matemàtic anglès James Joseph Sylvester el 1893. Ho va fer en forma de problema a una columna de la revista . Uns quaranta anys després, Tibor Gallai, un matemàtic hongarès, va demostrar el teorema. (ca) Der Satz von Sylvester-Gallai ist ein mathematischer Satz der ebenen Geometrie. Er besagt, dass zu jeder endlichen Menge von Punkten, die nicht alle auf einer Geraden, aber in einer Ebene liegen, eine Gerade existiert, die genau zwei Punkte der Menge enthält. Benannt ist er nach James Joseph Sylvester, der die Aussage 1893 in der Educational Times erstmals formulierte, und Tibor Gallai, der 1944 den ersten Beweis veröffentlichte, nachdem Paul Erdős das Problem 1943 neu stellte. Der erste bekannte Beweis stammt von Eberhard Melchior aus dem Jahre 1940. (de) في الهندسة الرياضية، تنص مبرهنة سلفستر غالاي (بالإنجليزية: Sylvester–Gallai theorem) على أنه في حال وجد مجموعة منتهية من النقاط في المستوي فإن هذه النقاط إما أن: 1. * تكون جميع النقاط واقعة على مستقيم واحد 2. * يوجد مستقيم يحتوي تماماً اثنتين من هذه النقاط. تم تقديم هذه المبرهنة من قبل جايمس جوزيف سلفستر في عام 1893، وبشكل منفصل من قبل بول إيردوس في عام 1943 ومن ثم برهنت من قبل في عام 1944. هذه المبرهنة لا تنطبق على مجموعة غير منتهية من النقاط. (ar) En géométrie discrète, le théorème de Sylvester-Gallai affirme qu'étant donné un ensemble fini de points du plan, on a l'alternative suivante : * soit tous les points sont alignés, * soit il existe une droite qui contient exactement deux de ces points. Les droites contenant exactement deux points sont nommées droites ordinaires. Ce théorème ne s'applique pas à des ensembles infinis de points : il suffit pour s'en convaincre de considérer l'ensemble des points de coordonnées entières dans le plan euclidien. (fr) O teorema de Sylvester–Gallai afirma que dado um número finito de pontos no plano euclidiano, das duas uma: 1. * todos os pontos são colineares (estão em linha reta); ou 2. * há uma reta que passa por exatamente dois dos pontos. Esta afirmação foi conjecturada por James Joseph Sylvester, e foi o primeiro a prová-la. Sem saber da prova de Melchior, Paul Erdős a enunciou como conjectura, que então foi provada por Tibor Gallai, e logo em seguida por outros. Uma demonstração encontrada por em 1948 foi bastante apreciada por Erdős. (pt) Теорема Сильвестра — классический результат комбинаторной геометрии о конфигурациях прямых на плоскости. (ru) Sylvesters sats är en matematisk sats uppkallad efter matematikern James Joseph Sylvester som lyder Om man har en ändlig punktmängd i planet, , där alla punkter inte ligger längs en linje så finns det en linje som skär exakt två punkter. (sv) Теорема Сильвестра — класичний результат комбінаторної геометрії про конфігурації прямих на площині. (uk) 西爾維斯特–高洛伊定理（Sylvester–Gallai theorem）說明若在平面上有有限數目的點，點的數目多於2，它們不是全部共線，有一條線上剛好有兩點，如果过任意两点的直线都必过第三点，则所有的点共线。 這個定理在無限點的情況並不成立，可以考慮格點。 (zh) The Sylvester–Gallai theorem in geometry states that every finite set of points in the Euclidean plane has a line that passes through exactly two of the points or a line that passes through all of them. It is named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944. (en) Il teorema di Sylvester–Gallai (in origine una congettura nota come problema di Sylvester) afferma che, dato un insieme finito di almeno 3 punti del piano, non è possibile disporli in una configurazione tale che ogni retta che passi per due punti ne contenga anche un terzo, fatto salvo il caso in cui siano tutti allineati. In altri termini, è vera la seguente alternativa: 1. * o tutti i punti sono allineati; 2. * oppure esiste almeno una retta che contiene solo due punti dell'insieme. (it) |
| rdfs:label | مبرهنة سلفستر-غالاي (ar) Teorema de Sylvester-Gallai (ca) Satz von Sylvester-Gallai (de) Théorème de Sylvester-Gallai (fr) Sylvester–Gallai theorem (en) Teorema di Sylvester-Gallai (it) Teorema de Sylvester–Gallai (pt) Теорема Сильвестра (ru) 西爾維斯特-高洛伊定理 (zh) Sylvesters sats (sv) Теорема Сильвестра — Галлаї (uk) |
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