{"id":1141,"date":"2021-12-30T23:25:23","date_gmt":"2021-12-30T21:25:23","guid":{"rendered":"http:\/\/www.joanillo.org\/?p=1141"},"modified":"2021-12-30T23:25:26","modified_gmt":"2021-12-30T21:25:26","slug":"geometria-per-acabar-lany","status":"publish","type":"post","link":"https:\/\/www.joanillo.org\/?p=1141","title":{"rendered":"Geometria per acabar l&#8217;any"},"content":{"rendered":"<p><a href=\"http:\/\/www.joanillo.org\/wp-content\/uploads\/2021\/12\/poligon_regular.png\"><img src=\"http:\/\/www.joanillo.org\/wp-content\/uploads\/2021\/12\/poligon_regular.png\" alt=\"\" title=\"v\u00e8rtex i diagonals en un pol\u00edgon regular\" width=\"276\" class=\"alignright size-medium wp-image-2338\"><\/a>Per acabar l&#8217;any, el repte de fer i entendre una demostraci\u00f3 de geometria.<br \/>\nEl problema: donat un <b>pol\u00edgon regular<\/b> de N costats (per exemple un hept\u00e0gon), demostrar que els angles que formen les diagonals que surten d&#8217;un v\u00e8rtex, tenen tots aquests angles el mateix valor, i dir quin valor \u00e9s.<\/p>\n<p>La demostraci\u00f3 del repte est\u00e0 en el <a href=\"http:\/\/www.joanillo.org\/wp-content\/uploads\/2021\/12\/demostracio_geometria_poligon_regular.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">pdf<\/a> de sota. No \u00e9s una demostraci\u00f3 evident, \u00e9s un d&#8217;aquells casos en qu\u00e8 l&#8217;enunciat \u00e9s molt f\u00e0cil, penses que la demostraci\u00f3 ser\u00e0 r\u00e0pida i f\u00e0cil, i despr\u00e9s no ho \u00e9s. En aquest cas, he hagut d&#8217;aprofundir en els <em>quadril\u00e0ters inscrits<\/em> en una circumfer\u00e8ncia (que tamb\u00e9 s&#8217;anomenen <em>c\u00edclics<\/em>), i les seves propietats: concretament, en un pol\u00edgon inscrit els angles oposats s\u00f3n <em>suplementaris<\/em> (sumen 180\u00ba).<\/p>\n<p>Per fer la demostraci\u00f3 \u00e9s necess\u00e0ria una cadena de resultats i demostracions pr\u00e8vies, algunes evidents, i d&#8217;altres no tant.<\/p>\n<p>Bon any <b>2022<\/b> i moltes matem\u00e0tiques per al nou any!<\/p>\n<p>La demostraci\u00f3: <a href=\"http:\/\/www.joanillo.org\/wp-content\/uploads\/2021\/12\/demostracio_geometria_poligon_regular.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">demostraci\u00f3<\/a> de qu\u00e8 en un pol\u00edgon regular les diagonals que parteixen d&#8217;un v\u00e8rtex tenen el mateix angle: 180\/N.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Per acabar l&#8217;any, el repte de fer i entendre una demostraci\u00f3 de geometria. El problema: donat un pol\u00edgon regular de N costats (per exemple un hept\u00e0gon), demostrar que els angles que formen les diagonals que surten d&#8217;un v\u00e8rtex, tenen tots aquests angles el mateix valor, i dir quin valor \u00e9s. La demostraci\u00f3 del repte est\u00e0 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[16,12,48],"tags":[],"_links":{"self":[{"href":"https:\/\/www.joanillo.org\/index.php?rest_route=\/wp\/v2\/posts\/1141"}],"collection":[{"href":"https:\/\/www.joanillo.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.joanillo.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.joanillo.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.joanillo.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1141"}],"version-history":[{"count":1,"href":"https:\/\/www.joanillo.org\/index.php?rest_route=\/wp\/v2\/posts\/1141\/revisions"}],"predecessor-version":[{"id":1142,"href":"https:\/\/www.joanillo.org\/index.php?rest_route=\/wp\/v2\/posts\/1141\/revisions\/1142"}],"wp:attachment":[{"href":"https:\/\/www.joanillo.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1141"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.joanillo.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1141"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.joanillo.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1141"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}