{"id":424,"date":"2017-02-14T21:28:35","date_gmt":"2017-02-14T20:28:35","guid":{"rendered":"http:\/\/www2.mathnique.com\/site\/?page_id=424"},"modified":"2018-03-30T15:37:20","modified_gmt":"2018-03-30T13:37:20","slug":"triangles","status":"publish","type":"page","link":"https:\/\/www.mathnique.com\/site\/triangles\/","title":{"rendered":"Triangles"},"content":{"rendered":"<ul>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1629 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/travaux3-300x252.png\" alt=\"\" width=\"112\" height=\"94\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/travaux3-300x252.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/03\/travaux3.png 718w\" sizes=\"auto, (max-width: 112px) 85vw, 112px\" \/><\/li>\n<li><strong><span style=\"color: #ff0000;\">In\u00e9galit\u00e9 triangulaire (La ligne droite est le plus court\u00a0chemin):<\/span><\/strong>\n<ul>\n<li>Sujet :\u00a0<a href=\"http:\/\/www2.mathnique.com\/site\/inegalite-triangulaire\/\" target=\"_blank\" rel=\"noopener noreferrer\">http:\/\/www2.mathnique.com\/site\/inegalite-triangulaire\/<\/a><\/li>\n<li>Corrig\u00e9 :<a href=\"http:\/\/www2.mathnique.com\/site\/corrige-pbs-sur-linegalite-triangulaire\/\" target=\"_blank\" rel=\"noopener noreferrer\">http:\/\/www2.mathnique.com\/site\/corrige-pbs-sur-linegalite-triangulaire\/<\/a><\/li>\n<li>Enigme : O\u00f9 placer $M$ sur la droite $(\\mathcal{D})$ pour que le chemin $AMB$ soit minimum ?<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2010 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/MAMBminimum-300x208.png\" alt=\"\" width=\"300\" height=\"208\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/MAMBminimum-300x208.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/MAMBminimum.png 393w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><\/li>\n<\/ul>\n<\/li>\n<li><strong><span style=\"color: #ff0000;\">Les points importants :<\/span><\/strong>\n<ul>\n<li><em><strong><span style=\"color: #ff0000;\">Le centre de gravit\u00e9<\/span><\/strong><\/em> \u00a0ou l'isobarycentre $G$ des points $A,B,C$ est le seul point v\u00e9rifiant l'\u00e9galit\u00e9 vectorielle suivante :<br \/>\n$\\overrightarrow{GA} +\u00a0\\overrightarrow{GB} +\u00a0\\overrightarrow{GC} = \\overrightarrow{0}$<br \/>\nEn notant $A'$ le milieu de $[BC]$ ,alors on a $\\overrightarrow{A'B} \u00a0+ \\overrightarrow{A'C} = \\overrightarrow{0}$ donc :<br \/>\n$\\overrightarrow{GA'} +\u00a0\u00a0\\overrightarrow{A'A} +\\overrightarrow{GA'} +\u00a0\\overrightarrow{A'B} + \\overrightarrow{GA'} + \\overrightarrow{A'C} = \\overrightarrow{0}$ d'o\u00f9<br \/>\n$3 \\overrightarrow{GA'} +\u00a0\\overrightarrow{A'A'} = \\overrightarrow{0}$<br \/>\n$\\overrightarrow{GA'} = \\dfrac{1}{3}\u00a0\\overrightarrow{AA'}$.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-2007 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/medianesABC-300x273.png\" alt=\"\" width=\"300\" height=\"273\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/medianesABC-300x273.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/medianesABC.png 328w\" sizes=\"auto, (max-width: 300px) 85vw, 300px\" \/><br \/>\n<strong>Par cons\u00e9quent, le centre de gravit\u00e9 $G$ est le point de concours des 3 m\u00e9dianes $(AA'),(BB'),(CC')$.<\/strong><br \/>\n<strong>Il est situ\u00e9 sur chaque m\u00e9diane au $\\dfrac{1}{3}$\u00a0de la base et \u00e0 $\\dfrac{2}{3}$ du sommet.<\/strong><\/li>\n<li><em><strong><span style=\"color: #ff0000;\">L'orthocentre<\/span><\/strong><\/em>\u00a0$H$ st le point de concours des hauteurs.<\/li>\n<li><em><strong><span style=\"color: #ff0000;\">Le centre du cercle circonscrit<\/span><\/strong><\/em>\u00a0$O$ est le point de concours des m\u00e9diatrices.<\/li>\n<li><em><strong><span style=\"color: #ff0000;\">Le centre du cercle inscrit<\/span><\/strong><\/em>\u00a0$I$ est le sprint de concours des bissectrices int\u00e9rieures.<\/li>\n<li><span style=\"color: #ff0000;\"><strong><em>$O,G$ et $H$ sont align\u00e9s<\/em><\/strong><\/span> sur une droite dite <strong>Droite d'EULER<\/strong> avec la relation suivante<br \/>\n$\\overrightarrow{OH} = 3\u00a0\\overrightarrow{OG}$<br \/>\nPour ma\u00eetriser ce th\u00e8me , je vous propose un sujet sur la Droite d'Euler:\u00a0<a href=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/Euler.pdf\">Euler<\/a><\/li>\n<\/ul>\n<\/li>\n<li><strong><span style=\"color: #ff0000;\">Thal\u00e8s et sa r\u00e9ciproque :<\/span><\/strong>\n<ul>\n<li><strong><em><span style=\"color: #ff0000;\">Thal\u00e8s Triangle et Thal\u00e8s Papillon:<br \/>\n<\/span><\/em><\/strong>Soit un triangle $ABC$.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\" wp-image-2025 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalestrianglepapillon-300x78.png\" alt=\"\" width=\"531\" height=\"138\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalestrianglepapillon-300x78.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalestrianglepapillon-768x200.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalestrianglepapillon-1024x267.png 1024w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalestrianglepapillon-1200x313.png 1200w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalestrianglepapillon.png 1626w\" sizes=\"auto, (max-width: 531px) 85vw, 531px\" \/><br \/>\n- Si $M$ est un point de la droite $(AB)$ diff\u00e9rent de $A$<br \/>\n- Si\u00a0$N$ est un point de la droite $(AC)$ diff\u00e9rent de $A$<br \/>\n- Si $(MN)$ et $(BC)$ sont parall\u00e8les<br \/>\nAlors $\\dfrac{AM}{AB} =\u00a0\\dfrac{AN}{AC} =\u00a0\\dfrac{MN}{BC} $<\/li>\n<li><em><strong><span style=\"color: #ff0000;\">Corollaire : Th\u00e9or\u00e8me 1 des milieux<\/span><\/strong><\/em><br \/>\nLa droite qui passe par le milieu d'un \u00a0c\u00f4t\u00e9 d'un triangle et qui est parall\u00e8le \u00e0 un autre c\u00f4t\u00e9 passe alors par le milieu du troisi\u00e8me c\u00f4t\u00e9.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\" wp-image-2026 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalesmilieux-300x279.png\" alt=\"\" width=\"133\" height=\"123\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalesmilieux-300x279.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalesmilieux.png 588w\" sizes=\"auto, (max-width: 133px) 85vw, 133px\" \/><\/li>\n<li><em><strong><span style=\"color: #ff0000;\">R\u00e9ciproque Thal\u00e8s triangle :<br \/>\n<\/span><\/strong><\/em>Dans le cas d'une des trois figures suivantes :<img loading=\"lazy\" decoding=\"async\" class=\" wp-image-2024 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalesreciproque-300x80.png\" alt=\"\" width=\"431\" height=\"115\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalesreciproque-300x80.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalesreciproque-768x205.png 768w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalesreciproque-1024x274.png 1024w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalesreciproque-1200x321.png 1200w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalesreciproque.png 1264w\" sizes=\"auto, (max-width: 431px) 85vw, 431px\" \/><br \/>\n<span style=\"color: #ff0000;\"><span style=\"color: #000000;\">Si\u00a0l'on a $\\dfrac{AM}{AB} =\u00a0\\dfrac{AN}{AC}$ \u00a0alors les droites\u00a0$(MN)$ et $(BC)$ sont parall\u00e8les.<\/span>\u00a0<\/span><\/li>\n<li><em><strong><span style=\"color: #ff0000;\">Corollaire : Th\u00e9or\u00e8me 2 des milieux<\/span><\/strong><\/em><br \/>\nLa droite qui joint les milieux de deux c\u00f4t\u00e9s d'un triangle est parall\u00e8le au troisi\u00e8me c\u00f4t\u00e9.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\" wp-image-2026 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalesmilieux-300x279.png\" alt=\"\" width=\"140\" height=\"131\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalesmilieux-300x279.png 300w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/thalesmilieux.png 588w\" sizes=\"auto, (max-width: 140px) 85vw, 140px\" \/><\/li>\n<\/ul>\n<\/li>\n<li><strong><span style=\"color: #ff0000;\">Pythagore et sa r\u00e9ciproque \u00a0:<br \/>\n<\/span><\/strong><strong><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-2018 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/pythagore-289x300.png\" alt=\"\" width=\"82\" height=\"85\" srcset=\"https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/pythagore-289x300.png 289w, https:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/pythagore.png 360w\" sizes=\"auto, (max-width: 82px) 85vw, 82px\" \/><\/strong><\/p>\n<ul>\n<li>Si un triangle $ABC$ est rectangle en $A$ alors $BC^2 = AB^2 + AC^2$<\/li>\n<li>Si dans un\u00a0triangle $ABC$ on a $BC^2 = AB^2 + AC^2$ alors $ABC$ est rectangle en $A$.<br \/>\n<strong><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-2017 aligncenter\" src=\"http:\/\/www.mathnique.com\/site\/wp-content\/uploads\/2017\/02\/pythagore1.png\" alt=\"\" width=\"210\" height=\"205\" \/><\/strong><\/li>\n<\/ul>\n<\/li>\n<li><strong><span style=\"color: #ff0000;\">Relations m\u00e9triques :<br \/>\n<\/span><\/strong><\/p>\n<ul>\n<li>dans un triangle rectangle<\/li>\n<li>dans un triangle quelconque<\/li>\n<\/ul>\n<\/li>\n<li><span style=\"color: #ff0000;\"><strong>Triangle \u00e9quilat\u00e9ral<\/strong><\/span><br \/>\n<em>\"Je suis all\u00e9 trop loin<\/em><br \/>\n<em>Avec mon souci d'ordre\u00a0<\/em><br \/>\n<em>Rien ne peut plus venir\"<\/em><br \/>\n<em>GUILLEVIC - Comptines euclidiennes - Po\u00e9sie - Gallimard 1967<\/em><\/p>\n<ul>\n<li>Hauteur dans un triangle \u00e9quilat\u00e9ral<\/li>\n<li>La somme des distances d'un point int\u00e9rieur aux 3 c\u00f4t\u00e9s est constant et vaut la valeur de la hauteur.<\/li>\n<\/ul>\n<\/li>\n<li><strong><span style=\"color: #ff0000;\">Triangle rectangle<\/span><\/strong>\n<ul>\n<li>Th\u00e9or\u00e8me de Pythagore<\/li>\n<li>Triangle inscrit dans un demi-cercle.<\/li>\n<\/ul>\n<\/li>\n<li><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>In\u00e9galit\u00e9 triangulaire (La ligne droite est le plus court\u00a0chemin): Sujet :\u00a0http:\/\/www2.mathnique.com\/site\/inegalite-triangulaire\/ Corrig\u00e9 :http:\/\/www2.mathnique.com\/site\/corrige-pbs-sur-linegalite-triangulaire\/ Enigme : O\u00f9 placer $M$ sur la droite $(\\mathcal{D})$ pour que le chemin $AMB$ soit minimum ? Les points importants : Le centre de gravit\u00e9 \u00a0ou l'isobarycentre $G$ des points $A,B,C$ est le seul point v\u00e9rifiant l'\u00e9galit\u00e9 vectorielle suivante : $\\overrightarrow{GA} +\u00a0\\overrightarrow{GB} &hellip; <a href=\"https:\/\/www.mathnique.com\/site\/triangles\/\" class=\"more-link\">Continuer la lecture<span class=\"screen-reader-text\"> de &laquo;&nbsp;Triangles&nbsp;&raquo;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-424","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/424","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/comments?post=424"}],"version-history":[{"count":22,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/424\/revisions"}],"predecessor-version":[{"id":2028,"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/pages\/424\/revisions\/2028"}],"wp:attachment":[{"href":"https:\/\/www.mathnique.com\/site\/wp-json\/wp\/v2\/media?parent=424"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}