{"id":15,"date":"2025-10-01T20:50:00","date_gmt":"2025-10-01T18:50:00","guid":{"rendered":"http:\/\/maths-chambon.fr\/?page_id=15"},"modified":"2026-01-14T16:39:09","modified_gmt":"2026-01-14T14:39:09","slug":"id-programmes-et-resultats","status":"publish","type":"page","link":"https:\/\/maths-chambon.fr\/?page_id=15","title":{"rendered":"Deux exercices"},"content":{"rendered":"<p><span style=\"color: #ff0000;\"><strong><span style=\"text-decoration: underline;\">Exercice : 1+1=3 ?<\/span>\u00a0 <\/strong><\/span><span style=\"color: #0000ff;\"><strong><a style=\"color: #0000ff;\" href=\"https:\/\/www.youtube.com\/watch?v=JDF2BVeFoZ8\">https:\/\/www.youtube.com\/watch?<\/a><\/strong><\/span><\/p>\n<p><span style=\"color: #ffffff;\">\u00a0 \u00a0<\/span><span style=\"color: #800000;\"><strong>Mais o\u00f9 donc est l&rsquo;erreur ?\u00a0 (Assez facile quand m\u00eame !)<\/strong><\/span><\/p>\n<p><span style=\"color: #ffffff;\">\u00a0<\/span><span style=\"color: #800000;\"><span style=\"color: #ffffff;\">&#8212;\u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span>&#8212;&#8212;&#8212;&#8212;<\/span><\/p>\n<p><strong><span style=\"color: #ff0000;\"><span style=\"text-decoration: underline;\">Autre exercice conduisant aux courbes et \u00e0 la g\u00e9om\u00e9trie du triangle (*)<\/span><\/span>\u00a0<span style=\"color: #800000;\">Ayant le polyn\u00f4me : P(x)=(x-a)(x-b)(x-c) de racines complexes a,b,c, affixes d\u2019un triangle (vu en Terminale) o\u00f9 sont situ\u00e9es les 2 racines de P\u2019(x) ?<\/span><\/strong><\/p>\n<p><span style=\"color: #800000;\">1) En exercice, pas trop difficile, on montre qu\u2019elles sont situ\u00e9es dans le triangle (<strong>Th\u00e9or\u00e8me de Lucas<\/strong>). <\/span><\/p>\n<p><span style=\"color: #800000;\">2) On peut aussi montrer (*) qu\u2019il y a une et une seule ellipse (<strong>&lsquo;de Steiner inscrite&rsquo;<\/strong>) d\u2019aire maximum inscrite dans le triangle; et ses \u00ab foyers \u00bb sont les points cherch\u00e9s ! Ci-dessous en <strong><span style=\"color: #008000;\">vert<\/span><\/strong>, elle est tangente au milieu des c\u00f4t\u00e9s.\u00a0 \u00a0\u00a0<\/span><\/p>\n<p><a href=\"http:\/\/maths-chambon.fr\/wp-content\/uploads\/2024\/10\/Ellipse-de-Steiner-bis.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-2964\" src=\"http:\/\/maths-chambon.fr\/wp-content\/uploads\/2024\/10\/Ellipse-de-Steiner-bis-150x150.jpg\" alt=\"\" width=\"117\" height=\"117\" \/><\/a><\/p>\n<p><strong><span style=\"color: #ff0000;\">Notes:<\/span><\/strong> <span style=\"color: #993300;\"><strong>1) Longueur et \u00ab courbure \u00bb des courbes sont vues dans le COURS de MATHS et le livre d&rsquo;exercices qui va avec. Ainsi que le lieu du \u00ab\u00a0centre de courbure\u00a0\u00bb dite<\/strong><\/span><span style=\"color: #993300;\"><strong> \u00ab\u00a0courbe d\u00e9velopp\u00e9e\u00a0\u00bb . La d\u00e9velopp\u00e9e est aussi l&rsquo;enveloppe des normales (Th\u00e9or\u00e8me).<\/strong><\/span><\/p>\n<p><span style=\"color: #993300;\"><strong> 2) Sur les coniques signalons, \u00e0 titre culturel, le c\u00e9l\u00e8bre \u00ab th\u00e9or\u00e8me de l\u2019hexagramme mystique \u00bb, trouv\u00e9 par Blaise Pascal \u00e0 l&rsquo;\u00e2ge de 16 ans, et dont il tirait plus de 400 corollaires ! \u2026<\/strong><\/span><\/p>\n<p><span style=\"color: #993300;\"><strong> 3) Ci-dessous, 2\u00e8me courbe : <span style=\"text-decoration: underline;\">la \u00ab chainette \u00bb,<\/span> d\u2019\u00e9quation y=[exp(x)+exp(-x)]\/2 qui a exactement la forme d\u2019une petite cha\u00eene port\u00e9e au cou (cf. \u00e9quations diff\u00e9rentielles). Mais \u00e0 ne pas confondre (comme Galil\u00e9e)<\/strong><\/span> <span style=\"color: #993300;\"><strong>avec <span style=\"text-decoration: underline;\">la parabole<\/span> ! <\/strong><\/span><br \/>\n<strong><span style=\"color: #0000ff;\"><span style=\"color: #800000;\"><span style=\"color: #0000ff;\">\u00ab\u00a0Astro\u00efde\u00a0\u00bb<\/span> et sa \u00ab\u00a0<span style=\"color: #ff0000;\">d\u00e9velopp\u00e9e<\/span>\u00a0\u00bb &#8230;\u00a0 <span style=\"color: #ff0000;\">et<\/span> <\/span><\/span> <\/strong> <a title=\"Chainette\" href=\"http:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/chainette1.JPG\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/chainette1.thumbnail.JPG\" alt=\"Chainette\" width=\"60\" height=\"59\" \/><\/a><span style=\"color: #0000ff;\"><a href=\"http:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Astroide.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-thumbnail wp-image-1854\" title=\"Astroide\" src=\"http:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Astroide-150x150.jpg\" alt=\"\" width=\"66\" height=\"66\" srcset=\"https:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Astroide-150x150.jpg 150w, https:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Astroide-300x300.jpg 300w, https:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Astroide.jpg 400w\" sizes=\"(max-width: 66px) 100vw, 66px\" \/><\/a><strong><span style=\"color: #ff0000;\"> la \u00ab\u00a0Chainette\u00a0\u00bb<\/span><\/strong><\/span><\/p>\n<p><strong><span style=\"color: #800000;\">Puis ellipse, cyclo\u00efde et cardio\u00efde <span style=\"color: #0000ff;\">en bleu<\/span> et les d\u00e9velopp\u00e9es <span style=\"color: #ff0000;\">en rouge<\/span><\/span><\/strong><a href=\"http:\/\/maths-chambon.fr\/wp-content\/uploads\/2019\/11\/dev_ellipse.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-thumbnail wp-image-2371\" src=\"http:\/\/maths-chambon.fr\/wp-content\/uploads\/2019\/11\/dev_ellipse-150x150.jpg\" alt=\"\" width=\"150\" height=\"150\" srcset=\"https:\/\/maths-chambon.fr\/wp-content\/uploads\/2019\/11\/dev_ellipse-150x150.jpg 150w, https:\/\/maths-chambon.fr\/wp-content\/uploads\/2019\/11\/dev_ellipse-300x300.jpg 300w, https:\/\/maths-chambon.fr\/wp-content\/uploads\/2019\/11\/dev_ellipse.jpg 400w\" sizes=\"(max-width: 150px) 100vw, 150px\" \/><\/a><a href=\"http:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Cycloide.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-thumbnail wp-image-1850\" src=\"http:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Cycloide-150x150.jpg\" alt=\"\" width=\"150\" height=\"150\" srcset=\"https:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Cycloide-150x150.jpg 150w, https:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Cycloide-300x300.jpg 300w, https:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Cycloide.jpg 400w\" sizes=\"(max-width: 150px) 100vw, 150px\" \/><\/a><a href=\"http:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Cardioide.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft size-thumbnail wp-image-1848\" src=\"http:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Cardioide-150x150.jpg\" alt=\"\" width=\"150\" height=\"150\" srcset=\"https:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Cardioide-150x150.jpg 150w, https:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Cardioide-300x300.jpg 300w, https:\/\/maths-chambon.fr\/wp-content\/uploads\/2007\/10\/Cardioide.jpg 400w\" sizes=\"(max-width: 150px) 100vw, 150px\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Exercice : 1+1=3 ?\u00a0 https:\/\/www.youtube.com\/watch? \u00a0 \u00a0Mais o\u00f9 donc est l&rsquo;erreur ?\u00a0 (Assez facile quand m\u00eame !) \u00a0&#8212;\u00a0 \u00a0 \u00a0 \u00a0 \u00a0&#8212;&#8212;&#8212;&#8212; Autre exercice conduisant aux courbes et \u00e0 la g\u00e9om\u00e9trie du triangle (*)\u00a0Ayant le polyn\u00f4me : P(x)=(x-a)(x-b)(x-c) de racines complexes a,b,c, affixes d\u2019un triangle (vu en Terminale) o\u00f9 sont situ\u00e9es les 2 racines de [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-15","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Deux exercices - Maths-pr\u00e9pas, L1-L3; Histoire locale (43) Henri Chambon<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/maths-chambon.fr\/?page_id=15\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Deux exercices - Maths-pr\u00e9pas, L1-L3; Histoire locale (43) Henri Chambon\" \/>\n<meta property=\"og:description\" content=\"Exercice : 1+1=3 ?\u00a0 https:\/\/www.youtube.com\/watch? 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