{"id":339,"date":"2023-11-26T11:04:38","date_gmt":"2023-11-26T10:04:38","guid":{"rendered":"https:\/\/gjmaths.fr\/?page_id=339"},"modified":"2023-11-26T11:04:42","modified_gmt":"2023-11-26T10:04:42","slug":"isogones","status":"publish","type":"page","link":"https:\/\/gjmaths.fr\/index.php\/isogones\/","title":{"rendered":"Isogones"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\" id=\"isogone\">Une <strong>courbe isogone d&rsquo;une conique<\/strong> (C) donn\u00e9e est telle que, de chacun de ses points, on peut mener deux tangentes \u00e0 la conique (C) d\u00e9terminant un angle g\u00e9om\u00e9trique constant. Parmi les isogones de (C), il peut exister (mais ce n&rsquo;est pas toujours le cas) la courbe orthoptique de la conique : de chacun des points de l&rsquo;orthoptique, on peut mener deux tangentes \u00e0 (C) perpendiculaires.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Cette page pr\u00e9sente quelques documents relatifs \u00e0 ces isogones. Il serait bien \u00e9tonnant que les g\u00e9om\u00e8tres des XVIII\u00e8me et XIX\u00e8me si\u00e8cle n&rsquo;aient pas d\u00e9j\u00e0 abondamment creus\u00e9 le sujet mais, pour ma part, je n&rsquo;ai pas trouv\u00e9 de r\u00e9f\u00e9rence pr\u00e9cise.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">L&rsquo;\u00e9tude de telles isogones donne lieu \u00e0 un r\u00e9investissement assez dense des techniques relatives au second degr\u00e9 (discriminant, signe du trin\u00f4me, liens entre coefficients et racines, sarabande d&rsquo;expressions bicarr\u00e9es&#8230;). Il est vivement conseill\u00e9 de se munir d&rsquo;un logiciel de calcul formel et d&rsquo;un autre de repr\u00e9sentations graphiques permettant de tracer des courbes d\u00e9finies par une relation f(x, y) = 0.<br>Ces documents ne sont pas destin\u00e9s prioritairement aux candidats au CAPES mais peuvent \u00e9ventuellement int\u00e9resser des \u00e9tudiants de classes pr\u00e9pa ou de Master de Math\u00e9matiques.<br>Ils se pr\u00e9sentent sous forme de divers probl\u00e8mes avec sujet et corrig\u00e9.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Courbes isogones d&rsquo;une parabole<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"http:\/\/gjmaths.fr\/wp-content\/uploads\/2023\/11\/orthop.pdf\">Etude d&rsquo;un cas particulier<\/a> . On choisit pour cette \u00e9tude une parabole particuli\u00e8re.<\/li>\n\n\n\n<li><a href=\"http:\/\/gjmaths.fr\/wp-content\/uploads\/2023\/11\/isogopara.pdf\">Etude du cas g\u00e9n\u00e9ral<\/a> . M\u00eame sujet, mais avec un param\u00e8tre en plus.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Courbes isogones d&rsquo;une ellipse<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">Sont-ce des ellipses ?<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"http:\/\/gjmaths.fr\/wp-content\/uploads\/2023\/11\/orthellipse.pdf\">Etude d&rsquo;un exemple<\/a> .<\/li>\n\n\n\n<li><a href=\"http:\/\/gjmaths.fr\/wp-content\/uploads\/2023\/11\/orthelli.pdf\">Etude du cas g\u00e9n\u00e9ral<\/a> . Un essai de g\u00e9n\u00e9ralisation &#8230;<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Courbes isogones d&rsquo;une hyperbole<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Un sujet sur les <a href=\"http:\/\/gjmaths.fr\/wp-content\/uploads\/2023\/11\/orthyper.pdf\">courbes isogones d&rsquo;une hyperbole<\/a>. Apr\u00e8s paraboles et ellipses, voici les hyperboles. Ce n&rsquo;est pas si diff\u00e9rent de ce qu&rsquo;il se passe pour les ellipses, c&rsquo;est m\u00eame, il me semble, plus facile.<\/li>\n\n\n\n<li>Un exemple d&rsquo;hyperbole particuli\u00e8re avec une approche un peu diff\u00e9rente :<a href=\"http:\/\/gjmaths.fr\/wp-content\/uploads\/2023\/11\/orthyperequi.pdf\"> Isogones de l&rsquo;hyperbole d&rsquo;\u00e9quation y=1\/x <\/a>. Cette fois-ci l&rsquo;hyperbole est repr\u00e9sentative d&rsquo;une fonction. Mais cela ne change rien, toujours des quartiques.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Une courbe isogone d&rsquo;une conique (C) donn\u00e9e est telle que, de chacun de ses points, on peut mener deux tangentes \u00e0 la conique (C) d\u00e9terminant un angle g\u00e9om\u00e9trique constant. Parmi les isogones de (C), il peut exister (mais ce n&rsquo;est pas toujours le cas) la courbe orthoptique de la conique : de chacun des points [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"ocean_post_layout":"","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"","ocean_second_sidebar":"","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"","ocean_custom_header_template":"","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"","ocean_menu_typo_font_family":"","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"default","ocean_disable_heading":"default","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"default","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"","footnotes":""},"class_list":["post-339","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/gjmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/339","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gjmaths.fr\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/gjmaths.fr\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/gjmaths.fr\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gjmaths.fr\/index.php\/wp-json\/wp\/v2\/comments?post=339"}],"version-history":[{"count":1,"href":"https:\/\/gjmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/339\/revisions"}],"predecessor-version":[{"id":347,"href":"https:\/\/gjmaths.fr\/index.php\/wp-json\/wp\/v2\/pages\/339\/revisions\/347"}],"wp:attachment":[{"href":"https:\/\/gjmaths.fr\/index.php\/wp-json\/wp\/v2\/media?parent=339"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}