{"id":1495,"date":"2016-08-16T22:02:00","date_gmt":"2016-08-16T21:02:00","guid":{"rendered":"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/?p=1495"},"modified":"2018-02-17T02:06:17","modified_gmt":"2018-02-17T01:06:17","slug":"configurations-et-symetrie-centrale","status":"publish","type":"post","link":"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/configurations-et-symetrie-centrale\/","title":{"rendered":"Configurations et sym\u00e9trie centrale"},"content":{"rendered":"<p>Soit ABCD un parall\u00e9logramme de centre O.<\/p>\n<ul>\n<li>est le point de [AD] tel que ED=(1\/4) AD ;<\/li>\n<li>est le point de [BC] tel que\u00a0BF=(1\/4) BC.<\/li>\n<\/ul>\n<p>La droite (EF) coupe la droite (CD) en M et la droite (AB) en N.<\/p>\n<p>On se propose de d\u00e9montrer que le point O est le milieu du segment [EF]. Voici deux id\u00e9es de d\u00e9monstration :<\/p>\n<ol>\n<li><strong>a)<\/strong> Eric voudrait utiliser la configuration de Thal\u00e8s et passe en revue les hypoth\u00e8ses n\u00e9cessaires :<\/li>\n<\/ol>\n<ul>\n<li>les droites (ED) et (BF) sont parall\u00e8les ;<\/li>\n<li>les points B, O et D sont align\u00e9s ;<\/li>\n<li>les points E, O et F sont align\u00e9s.<\/li>\n<\/ul>\n<p>Il ne peut continuer \u2026 Pourquoi ?<\/p>\n<ol>\n<li><strong>b)<\/strong> Paul d\u00e9montre d\u2019abord que le quadrilat\u00e8re BEDF est un parall\u00e9logramme et conclut.<\/li>\n<\/ol>\n<p>R\u00e9diger la d\u00e9monstration de Paul.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>2.<\/strong> On se propose de d\u00e9montrer que le point O est aussi le milieu de [MN]. Lucile se sert de la sym\u00e9trie de centre O. Sa d\u00e9monstration d\u00e9bute ainsi :<\/p>\n<p>\u00ab Je cherche \u00e0 d\u00e9montrer que le point M est le sym\u00e9trique du point N par rapport \u00e0 O.<\/p>\n<p>Le point O est le centre du parall\u00e9logramme ABCD alors, par la sym\u00e9trie <em>s<\/em> de centre O, le point A a pour image C et le point B a pour image D ; la droite (AB) a donc pour image la droite (CD).<\/p>\n<p>Comme le point N est sur la droite (AB), son image par <em>s<\/em> est sur la droite (CD). \u00bb<\/p>\n<p>Terminer la d\u00e9monstration de Lucile.<\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-content\/uploads\/sites\/3\/2016\/08\/utiliser_configuration_symetrie_centrale.docx\"> Fiche \u00e9l\u00e8ve Configurations et sym\u00e9trie centrale word<\/a><\/p>\n<p><a href=\"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-content\/uploads\/sites\/3\/2016\/08\/utiliser_configuration_symetrie_centrale.pdf\">Fiche \u00e9l\u00e8ve Configurations et sym\u00e9trie centrale pdf<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Soit ABCD un parall\u00e9logramme de centre O. est le point de [AD] tel que ED=(1\/4) AD ; est le point de [BC] tel que\u00a0BF=(1\/4) BC. La droite (EF) coupe la droite (CD) en M et la droite (AB) en N. On se propose de d\u00e9montrer que le point O est le milieu du segment [EF]. &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/configurations-et-symetrie-centrale\/\">Lire la suite<\/a><\/p>\n","protected":false},"author":351,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_bbp_topic_count":0,"_bbp_reply_count":0,"_bbp_total_topic_count":0,"_bbp_total_reply_count":0,"_bbp_voice_count":0,"_bbp_anonymous_reply_count":0,"_bbp_topic_count_hidden":0,"_bbp_reply_count_hidden":0,"_bbp_forum_subforum_count":0,"footnotes":""},"categories":[23,22],"tags":[],"class_list":["post-1495","post","type-post","status-publish","format-standard","hentry","category-seconde","category-lycee","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-json\/wp\/v2\/posts\/1495","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-json\/wp\/v2\/users\/351"}],"replies":[{"embeddable":true,"href":"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-json\/wp\/v2\/comments?post=1495"}],"version-history":[{"count":3,"href":"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-json\/wp\/v2\/posts\/1495\/revisions"}],"predecessor-version":[{"id":1501,"href":"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-json\/wp\/v2\/posts\/1495\/revisions\/1501"}],"wp:attachment":[{"href":"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-json\/wp\/v2\/media?parent=1495"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-json\/wp\/v2\/categories?post=1495"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ent2d.ac-bordeaux.fr\/disciplines\/mathematiques\/wp-json\/wp\/v2\/tags?post=1495"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}