{"id":147440,"date":"2025-10-30T12:06:58","date_gmt":"2025-10-30T15:06:58","guid":{"rendered":"https:\/\/vestibulares.estrategia.com\/portal\/?p=147440"},"modified":"2026-02-02T18:19:39","modified_gmt":"2026-02-02T21:19:39","slug":"teorema-de-ptolomeu-a-relacao-entre-lados-e-diagonais-de-quadrilateros-ciclicos","status":"publish","type":"post","link":"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/teorema-de-ptolomeu-a-relacao-entre-lados-e-diagonais-de-quadrilateros-ciclicos\/","title":{"rendered":"Teorema de Ptolomeu: a rela\u00e7\u00e3o entre lados e diagonais de quadril\u00e1teros c\u00edclicos"},"content":{"rendered":"<p class=\"wp-block-paragraph\">Na <span id=\"urn:enhancement-37248d3b-eacf-441e-98a3-bcc39c358d8a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">geometria<\/span> existem diversos <span id=\"urn:enhancement-c18f38eb-dc7b-4326-a84c-dc53a6d55702\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">teoremas<\/span> que funcionam como ferramenta b&aacute;sica na resolu&ccedil;&atilde;o de problemas, contudo, alguns deles, como o <span id=\"urn:local-annotation-813655\" class=\"textannotation disambiguated\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema-de-ptolomeu\">Teorema de Ptolomeu<\/span>, servem para solucionar problemas um pouco mais complexos. Ele estabelece uma rela&ccedil;&atilde;o elegante entre os quatro lados e as duas diagonais de <span id=\"urn:local-annotation-192689\" class=\"textannotation disambiguated\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilateros-ciclicos\">quadril&aacute;teros c&iacute;clicos<\/span>. Entenda mais sobre o assunto no artigo do EV a seguir.<p class=\"wp-block-paragraph\"><div class=\"cta-medio\" style=\"background-image: url(https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2024\/08\/curso-enem-1-estrategia-vestibulares-ctamedio.jpg);\"><h3>Pacote Enem TOP <\/h3> <p>Estude com o Estrat\u00e9gia Vestibulares<\/p><div class=\"cta-botao\"><a id=\"cta-medio\" href=\"https:\/\/vestibulares.estrategia.com\/curso\/enem-top\/\" target=\"_blank\" style=\"background:rgb(255,150,0)\">Saiba mais<\/a><\/div><\/div>\n\n\n\n<\/p><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_86 counter-hierarchy ez-toc-counter ez-toc-transparent ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">Navegue pelo conte\u00fado<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/teorema-de-ptolomeu-a-relacao-entre-lados-e-diagonais-de-quadrilateros-ciclicos\/#O-que-e-um-quadrilatero-ciclico\" >O que &eacute; um quadril&aacute;tero c&iacute;clico<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/teorema-de-ptolomeu-a-relacao-entre-lados-e-diagonais-de-quadrilateros-ciclicos\/#O-Teorema-de-Ptolomeu\" >O Teorema de Ptolomeu<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/teorema-de-ptolomeu-a-relacao-entre-lados-e-diagonais-de-quadrilateros-ciclicos\/#A-reciproca-do-teorema\" >A rec&iacute;proca do teorema<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/teorema-de-ptolomeu-a-relacao-entre-lados-e-diagonais-de-quadrilateros-ciclicos\/#Demonstracao-do-Teorema-de-Ptolomeu\" >Demonstra&ccedil;&atilde;o do Teorema de Ptolomeu<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/teorema-de-ptolomeu-a-relacao-entre-lados-e-diagonais-de-quadrilateros-ciclicos\/#A-aplicacao-do-teorema\" >A aplica&ccedil;&atilde;o do teorema<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/teorema-de-ptolomeu-a-relacao-entre-lados-e-diagonais-de-quadrilateros-ciclicos\/#Questao-sobre-Teorema-de-Ptolomeu\" >Quest&atilde;o sobre Teorema de Ptolomeu<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/vestibulares.estrategia.com\/portal\/materias\/matematica\/teorema-de-ptolomeu-a-relacao-entre-lados-e-diagonais-de-quadrilateros-ciclicos\/#Consiga-a-aprovacao-com-o-Estrategia-Vestibulares\" >Consiga a aprova&ccedil;&atilde;o com o Estrat&eacute;gia Vestibulares!<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\" id=\"h-o-que-e-um-quadrilatero-ciclico\"><span class=\"ez-toc-section\" id=\"O-que-e-um-quadrilatero-ciclico\"><\/span>O que &eacute; um quadril&aacute;tero c&iacute;clico<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Para melhor compreender o <span id=\"urn:enhancement-c966d1cb-6460-412d-8d9d-767fc4406c73\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">teorema<\/span>, &eacute; necess&aacute;rio saber que ele &eacute; aplicado a <span id=\"urn:enhancement-ed37b576-5cff-4e6d-99bb-d9faae3d72fd\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;teros<\/span> c&iacute;clicos (ou inscrit&iacute;veis). De modo geral, um <span id=\"urn:enhancement-6f790c65-018c-4b40-b0c5-0383003bb365\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> &eacute; dito c&iacute;clico quando seus quatro v&eacute;rtices pertencem, ao mesmo tempo, a uma &uacute;nica <span id=\"urn:enhancement-83c1361b-a1a8-4762-baf8-ce14dacf4cf0\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span>. Em outras palavras, &eacute; poss&iacute;vel tra&ccedil;ar uma <span id=\"urn:enhancement-67a49aab-5845-4539-8e7c-056f5ddbdd87\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span> que passe por todos os quatro pontos que definem o <span id=\"urn:enhancement-c1607b53-f0bc-4eca-8a6b-0343bee1be9b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span>.<\/p><p class=\"wp-block-paragraph\">Uma propriedade fundamental dos <span id=\"urn:enhancement-3e2a5d58-4013-4aea-933d-bf9cadcf0766\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;teros<\/span> inscrit&iacute;veis &eacute; a rela&ccedil;&atilde;o entre seus <span id=\"urn:enhancement-18db93db-8cc8-4d3b-a1e4-43c1240c6655\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulos<\/span> internos opostos. Em todo <span id=\"urn:enhancement-50e71cb3-1092-41bb-894a-17451e32c50c\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> c&iacute;clico, a <span id=\"urn:enhancement-ed49d994-49d6-4f90-8425-4eed6d57e4d2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> dos <span id=\"urn:enhancement-85deeef7-2a93-4e06-86b1-11f4627a78cf\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulos<\/span> internos opostos &eacute; igual a 180&deg;, ou seja, s&atilde;o suplementares.<\/p><p class=\"wp-block-paragraph\">Seja <em>ABCD <\/em>um <span id=\"urn:enhancement-10e83cb0-b5df-441f-aaf9-8f0337ebff51\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> c&iacute;clico:<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"300\" height=\"299\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-104.png\" alt=\"\" class=\"wp-image-147460\" srcset=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-104.png 300w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-104-150x150.png 150w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/figure><p class=\"wp-block-paragraph\">Temos que:&nbsp;<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"196\" height=\"102\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-105.png\" alt=\"\" class=\"wp-image-147461\"><\/figure><p class=\"wp-block-paragraph\">Essa propriedade &eacute; uma consequ&ecirc;ncia direta do <strong><span id=\"urn:enhancement-89d40a70-14ef-45fb-aac8-0af6f15e5b3d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">Teorema<\/span> do <span id=\"urn:enhancement-10269f18-c05d-4a20-ac88-8d5b084cde68\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&Acirc;ngulo<\/span> Inscrito<\/strong>, que afirma que um <span id=\"urn:enhancement-fa1d827e-6137-4556-8da4-a831d0d38958\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> inscrito em uma <span id=\"urn:enhancement-5a80fe36-5b79-4b0a-b68b-89a5bba2809a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span> mede <strong>metade do arco<\/strong> que ele intercepta. Assim, o <span id=\"urn:enhancement-5e88758f-6b38-456d-afd7-3da409546305\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> &ang;A intercepta o arco BCD, enquanto o <span id=\"urn:enhancement-61ed53ee-0a8c-4ab2-ab99-24811f74416d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulo<\/span> &ang;C intercepta o arco complementar DAB.&nbsp;<\/p><p class=\"wp-block-paragraph\">Como esses dois arcos juntos formam a <span id=\"urn:enhancement-a4c6326a-ffe1-4fcd-a4e4-c42f28722e58\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span> completa &mdash; ou seja, correspondem a 360&deg; &mdash; a <span id=\"urn:enhancement-77388cec-240e-4c8b-b097-8312107df76c\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> das medidas dos <span id=\"urn:enhancement-7aec9b05-1d28-4693-8d50-08981c5a90f8\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulos<\/span> &ang;A e &ang;C ser&aacute; metade desse valor, que &eacute; 180&deg;.<\/p><p class=\"wp-block-paragraph\">Temos ainda que a rec&iacute;proca tamb&eacute;m &eacute; v&aacute;lida, se um <span id=\"urn:enhancement-c538066d-b5e2-451f-8c70-b43f2f13b93e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> convexo &ndash; <span id=\"urn:enhancement-5f54e95e-3615-4975-8233-2a43da9638b5\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/poligono\">pol&iacute;gono<\/span> que possui todos os <span id=\"urn:enhancement-040e5e39-0e78-4338-9dc0-9dd6cfb80971\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulos<\/span> internos menores que 180&deg; &ndash;&nbsp; possui <span id=\"urn:enhancement-6d0b2c59-4f55-4c6f-9ec6-1878d07b3b39\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulos<\/span> opostos que somam 180&deg;, ent&atilde;o ele &eacute; um <span id=\"urn:enhancement-47073c97-007e-45e0-9ba3-afd25c261dc3\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> c&iacute;clico.<\/p><h3 class=\"wp-block-heading\" id=\"h-exemplos-notaveis\">Exemplos Not&aacute;veis:<\/h3><ul class=\"wp-block-list\">\n<li><strong>Quadrado:<\/strong> todos os &acirc;ngulos internos medem 90&deg;. Como 90&deg; + 90&deg; = 180&deg;, todo quadrado &eacute; c&iacute;clico;<\/li>\n\n\n\n<li><strong>Ret&acirc;ngulo:<\/strong> similar ao quadrado, todos os &acirc;ngulos internos medem 90&deg;. A soma dos opostos &eacute; 180&deg;, logo, todo ret&acirc;ngulo &eacute; c&iacute;clico;<\/li>\n\n\n\n<li><strong>Trap&eacute;zio Is&oacute;sceles:<\/strong> os &acirc;ngulos de uma mesma base s&atilde;o congruentes, e os &acirc;ngulos adjacentes a um lado n&atilde;o paralelo s&atilde;o suplementares. Se os &acirc;ngulos da base maior s&atilde;o A e B, e os da menor s&atilde;o C e D, temos A=B e C=D. Como A+D = 180&deg; e C=D, logo,&nbsp; A+C = 180&deg;. Portanto, todo trap&eacute;zio is&oacute;sceles &eacute; c&iacute;clico.<\/li>\n<\/ul><h2 class=\"wp-block-heading\" id=\"h-o-teorema-de-ptolomeu\"><span class=\"ez-toc-section\" id=\"O-Teorema-de-Ptolomeu\"><\/span>O Teorema de Ptolomeu<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Com a defini&ccedil;&atilde;o de <span id=\"urn:enhancement-e550188c-ca06-46a8-b0e1-db79f619837f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> c&iacute;clico estabelecida, podemos enunciar o <span id=\"urn:enhancement-c78510c9-b6d9-49c8-9b61-07c558554544\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">teorema<\/span>:<\/p><p class=\"wp-block-paragraph\">&ldquo;Para um <span id=\"urn:enhancement-4cf1b460-4b85-4205-b5b8-bb13dabee71a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> c&iacute;clico ABCD, o produto das medidas de suas diagonais &eacute; igual &agrave; <span id=\"urn:enhancement-688fba73-114f-41ae-bff4-54b3c269950e\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> dos produtos das medidas dos lados opostos.&rdquo;<\/p><p class=\"wp-block-paragraph\">Definido um <span id=\"urn:enhancement-b28d87fa-964e-4545-b5b3-6a7fa7af0208\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> c&iacute;clico ABCD, com lados AB, BC, CD e DA e diagonais AC e BD, temos que:&nbsp;<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"358\" height=\"33\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-106.png\" alt=\"\" class=\"wp-image-147462\"><\/figure><p class=\"wp-block-paragraph\">A equa&ccedil;&atilde;o mostra a rela&ccedil;&atilde;o direta entre os segmentos que formam um <span id=\"urn:enhancement-56d9e238-12f1-413c-8045-d28ef64f4aa5\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> c&iacute;clico (os quatro lados e as duas diagonais).<\/p><h2 class=\"wp-block-heading\" id=\"h-a-reciproca-do-teorema\"><span class=\"ez-toc-section\" id=\"A-reciproca-do-teorema\"><\/span>A rec&iacute;proca do teorema<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Assim como a propriedade dos <span id=\"urn:enhancement-e8b7a388-83e8-4429-bb2e-12d4bac82763\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulos<\/span> opostos em <span id=\"urn:enhancement-feb9fbe4-1693-47b4-8d18-9e4a4c9b96c7\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;teros<\/span> c&iacute;clicos, o <strong><span id=\"urn:enhancement-00f46a6b-31b5-4d71-97e0-423330ed0f56\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">Teorema<\/span> de Ptolomeu<\/strong> possui uma <strong>rec&iacute;proca<\/strong> muito &uacute;til &mdash; conhecida como a <strong>Desigualdade de Ptolomeu<\/strong>. Essa desigualdade torna-se uma <strong>igualdade exata<\/strong> quando o <span id=\"urn:enhancement-b6fbdbd4-8fb8-4d83-a01f-92aeb52c6cc5\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> &eacute; c&iacute;clico.<\/p><p class=\"wp-block-paragraph\">De <span id=\"urn:enhancement-468cba8e-41e4-4ec5-8760-b0dfcd5eee0d\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">forma<\/span> direta, a rec&iacute;proca diz o seguinte:<\/p><p class=\"wp-block-paragraph\">&ldquo;Se em um <span id=\"urn:enhancement-e7227818-c683-4a86-952b-d16526a85858\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> convexo ABCD, o produto das medidas de suas diagonais for igual &agrave; <span id=\"urn:enhancement-51169680-445f-4cd0-9cbc-3dde056814d8\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> dos produtos das medidas dos lados opostos, ou seja, AC * BD = (AB * CD) + (BC * DA), ent&atilde;o o <span id=\"urn:enhancement-4a4a447b-48de-44f3-9a87-7228a4fd21d3\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> ABCD &eacute; c&iacute;clico.&rdquo;<\/p><p class=\"wp-block-paragraph\">Essa rec&iacute;proca &eacute; especialmente &uacute;til em problemas de <strong><span id=\"urn:enhancement-4b4cbe0d-7864-4ae5-af14-41cdbafccca6\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">geometria<\/span> avan&ccedil;ada<\/strong>, <strong>vestibulares <\/strong>e olimp&iacute;adas, pois permite identificar a presen&ccedil;a de uma <span id=\"urn:enhancement-7386cee8-70f0-4b4b-82f8-a7723e8f8f9b\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span> circunscrita sem precisar desenh&aacute;-la ou medir <span id=\"urn:enhancement-2077a6e7-a525-408e-8afa-4f11bd9cc0af\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulos<\/span><\/p><h2 class=\"wp-block-heading\" id=\"h-demonstracao-do-teorema-de-ptolomeu\"><span class=\"ez-toc-section\" id=\"Demonstracao-do-Teorema-de-Ptolomeu\"><\/span>Demonstra&ccedil;&atilde;o do Teorema de Ptolomeu<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Existem v&aacute;rias demonstra&ccedil;&otilde;es para o <span id=\"urn:enhancement-2f287d4f-43e2-4911-8678-e39764211f3f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">Teorema<\/span> de Ptolomeu (usando <span id=\"urn:enhancement-7eb675e8-f85e-4f46-8c49-07136b2e3f79\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/numeros-complexos\">n&uacute;meros complexos<\/span>, invers&atilde;o <span id=\"urn:enhancement-d39e25b1-50a5-47fb-adad-10d9f809ab06\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria\">geom&eacute;trica<\/span>, etc.), mas a demonstra&ccedil;&atilde;o cl&aacute;ssica utiliza apenas <span id=\"urn:enhancement-c16c256b-63f1-4d1f-a9e7-066ac672e119\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/geometria-euclidiana\">Geometria Euclidiana<\/span> e semelhan&ccedil;a de tri&acirc;ngulos. Veja os passos:<\/p><p class=\"wp-block-paragraph\">Dado um <span id=\"urn:enhancement-15c10e98-18f1-42e8-bdd2-a1948d2a86de\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> c&iacute;clico ABCD.<\/p><p class=\"wp-block-paragraph\"><strong>Constru&ccedil;&atilde;o auxiliar:<\/strong> Tra&ccedil;amos a diagonal AC. Constru&iacute;mos um ponto E sobre o segmento AC tal que ele satisfa&ccedil;a a condi&ccedil;&atilde;o &ang;CBE = &ang;ABD.<\/p><p class=\"wp-block-paragraph\"><strong>Primeira Semelhan&ccedil;a de Tri&acirc;ngulos<\/strong> Vamos analisar o tri&acirc;ngulo ABD e o tri&acirc;ngulo EBC.<\/p><ol class=\"wp-block-list\">\n<li>&ang;ABD = &ang;EBC (por constru&ccedil;&atilde;o).<\/li>\n\n\n\n<li>&ang;ADB =&ang;ECB (s&atilde;o &acirc;ngulos inscritos que enxergam o mesmo arco AB na circunfer&ecirc;ncia).<\/li>\n<\/ol><p class=\"wp-block-paragraph\">Como possuem dois <span id=\"urn:enhancement-e121ac5f-faea-42ff-843f-08d0034a9741\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&acirc;ngulos<\/span> congruentes (caso <span id=\"urn:enhancement-320731d8-2163-4b07-bcac-bcf1e9cd2ba6\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&Acirc;ngulo<\/span>&ndash;<span id=\"urn:enhancement-c988e2d7-bff2-489c-897d-3d8ef6a5ff52\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&Acirc;ngulo<\/span>), os tri&acirc;ngulos s&atilde;o semelhantes:<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"156\" height=\"19\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-107.png\" alt=\"\" class=\"wp-image-147463\" srcset=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-107.png 156w, https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-107-150x19.png 150w\" sizes=\"(max-width: 156px) 100vw, 156px\" \/><\/figure><p class=\"wp-block-paragraph\">Da semelhan&ccedil;a, podemos montar a <span id=\"urn:enhancement-c74bcdd5-b648-48a8-ad30-c313340357a0\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/proporcao\">propor&ccedil;&atilde;o<\/span> entre os lados hom&oacute;logos:<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"109\" height=\"55\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-108.png\" alt=\"\" class=\"wp-image-147464\"><\/figure><p class=\"wp-block-paragraph\">Multiplicando em cruz, obtemos nossa primeira rela&ccedil;&atilde;o:<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"214\" height=\"27\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-109.png\" alt=\"\" class=\"wp-image-147465\"><\/figure><p class=\"wp-block-paragraph\">Novamente, pelo caso <span id=\"urn:enhancement-14fee272-5f23-4c27-b213-878b4dda8741\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&Acirc;ngulo<\/span>&ndash;<span id=\"urn:enhancement-b3965822-7b6e-40e7-858b-b45c75a5ecdd\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/angulo\">&Acirc;ngulo<\/span>, os tri&acirc;ngulos s&atilde;o semelhantes:<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"172\" height=\"19\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-110.png\" alt=\"\" class=\"wp-image-147466\"><\/figure><p class=\"wp-block-paragraph\">Montando a <span id=\"urn:enhancement-afde77e5-8160-4b8e-99e1-682595e59d70\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/proporcao\">propor&ccedil;&atilde;o<\/span>:<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"114\" height=\"51\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-111.png\" alt=\"\" class=\"wp-image-147467\"><\/figure><p class=\"wp-block-paragraph\">Multiplicando em cruz, obtemos nossa segunda rela&ccedil;&atilde;o:<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"204\" height=\"27\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-112.png\" alt=\"\" class=\"wp-image-147468\"><\/figure><p class=\"wp-block-paragraph\"><strong>A <span id=\"urn:enhancement-244f23b7-9068-4310-b751-672031e95b1f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">Soma<\/span> Alg&eacute;brica:<\/strong> Agora, somamos as duas equa&ccedil;&otilde;es que encontramos: (Equa&ccedil;&atilde;o 1) + (Equa&ccedil;&atilde;o 2):<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"489\" height=\"31\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-113.png\" alt=\"\" class=\"wp-image-147469\"><\/figure><p class=\"wp-block-paragraph\">Colocamos BD em evid&ecirc;ncia:<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"417\" height=\"26\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-114.png\" alt=\"\" class=\"wp-image-147470\"><\/figure><p class=\"wp-block-paragraph\">Observe a constru&ccedil;&atilde;o: o ponto E est&aacute; sobre o segmento AC. Portanto, a <span id=\"urn:enhancement-5318c06c-9024-4da6-bcde-b4205c32d7d9\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/adicao\">soma<\/span> EC + AE &eacute; exatamente o comprimento da diagonal AC.<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"149\" height=\"19\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-115.png\" alt=\"\" class=\"wp-image-147471\"><\/figure><p class=\"wp-block-paragraph\">Substituindo na equa&ccedil;&atilde;o:<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"350\" height=\"32\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-116.png\" alt=\"\" class=\"wp-image-147472\"><\/figure><p class=\"wp-block-paragraph\">Rearranjando os termos para o formato padr&atilde;o, chegamos ao <span id=\"urn:enhancement-56791eda-d2e3-44f3-87b7-c01cae1e4c17\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">Teorema<\/span> de Ptolomeu:<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"348\" height=\"31\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-117.png\" alt=\"\" class=\"wp-image-147473\"><\/figure><h2 class=\"wp-block-heading\" id=\"h-a-aplicacao-do-teorema\"><span class=\"ez-toc-section\" id=\"A-aplicacao-do-teorema\"><\/span>A aplica&ccedil;&atilde;o do teorema<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Para conseguir fazer a aplica&ccedil;&atilde;o do <span id=\"urn:enhancement-04f64e04-a4f2-46f7-99db-762c8b03e630\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">Teorema<\/span> de Ptolomeu, &eacute; preciso identificar certos cen&aacute;rios espec&iacute;ficos. Como por exemplo:<\/p><ul class=\"wp-block-list\">\n<li><strong>Identifica&ccedil;&atilde;o do Quadril&aacute;tero C&iacute;clico:<\/strong> o problema deve afirmar que o quadril&aacute;tero &eacute; c&iacute;clico, ou fornecer informa&ccedil;&otilde;es que permitam concluir isso (&acirc;ngulos opostos somam 180&deg;, ou &eacute; um ret&acirc;ngulo, quadrado, trap&eacute;zio is&oacute;sceles);<\/li>\n\n\n\n<li><strong>Problema de Medida Desconhecida:<\/strong> o uso mais comum &eacute; quando o problema fornece 5 das 6 medidas (lados e diagonais) e pede a sexta; e<\/li>\n\n\n\n<li><strong>Problema de Prova (Rec&iacute;proca):<\/strong> o problema fornece todas as 6 medidas e pede para <em>provar<\/em> que o quadril&aacute;tero &eacute; c&iacute;clico.<\/li>\n<\/ul><p class=\"wp-block-paragraph\">Com o <span id=\"urn:enhancement-22b96dba-4ddc-4d6f-aee3-8bb05a6f4451\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">teorema<\/span> de Ptolomeu &eacute; poss&iacute;vel tamb&eacute;m resolver problemas diretos ou inversos (determinando uma diagonal a partir dos lados, ou um lado a partir das diagonais), al&eacute;m de verificar se um <span id=\"urn:enhancement-3555736e-430f-4935-86f9-05f4d09f69b7\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> &eacute; c&iacute;clico por meio de sua rec&iacute;proca.&nbsp;<\/p><p class=\"wp-block-paragraph\">Al&eacute;m disso, o <span id=\"urn:enhancement-e1f540fc-b6d1-436b-a0a5-8896778660af\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">teorema<\/span> pode ser combinado com outros resultados, como a Lei dos Cossenos e propriedades de tri&acirc;ngulos ou <span id=\"urn:enhancement-12424d3c-7cdb-405d-9438-e2293cecedde\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;teros<\/span> especiais, tornando-se uma estrat&eacute;gia vers&aacute;til em quest&otilde;es mais elaboradas de vestibulares.<\/p><h3 class=\"wp-block-heading\" id=\"h-caso-especial\">Caso especial<\/h3><p class=\"wp-block-paragraph\">Um <span id=\"urn:enhancement-2b8dcf7d-1fd8-4417-97a5-05e36bc9cd2a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/retangulo\">ret&acirc;ngulo<\/span> &eacute; um <span id=\"urn:enhancement-9842b256-2c70-474a-af43-47c08ca188c2\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/quadrilatero\">quadril&aacute;tero<\/span> c&iacute;clico. Seja um <span id=\"urn:enhancement-a0e8acaa-eed5-4739-8db2-57c0d64a9918\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/retangulo\">ret&acirc;ngulo<\/span> de lados <em>a<\/em> e <em>b <\/em>(AB = CD = a e BC = DA = b) e diagonal <em>d <\/em>(AC = BD = d). Aplicando o <span id=\"urn:enhancement-526178cb-a5e6-477f-b9bc-0878d285755f\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">teorema<\/span> de Ptolomeu, temos que:&nbsp;<\/p><figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"371\" height=\"150\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-118.png\" alt=\"\" class=\"wp-image-147474\"><\/figure><p class=\"wp-block-paragraph\">Ou seja, o <span id=\"urn:enhancement-5e470a24-fbc5-48d2-9a00-d6bb37b6640a\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/teorema\">Teorema<\/span> de Ptolomeu, quando aplicado a um <span id=\"urn:enhancement-79fa8b45-b304-4d3f-a6e3-d049cb5c7ea1\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/retangulo\">ret&acirc;ngulo<\/span>, reduz-se exatamente ao Teorema de Pit&aacute;goras.<\/p><h2 class=\"wp-block-heading\" id=\"h-questao-sobre-teorema-de-ptolomeu\"><span class=\"ez-toc-section\" id=\"Questao-sobre-Teorema-de-Ptolomeu\"><\/span>Quest&atilde;o sobre Teorema de Ptolomeu<span class=\"ez-toc-section-end\"><\/span><\/h2><h3 class=\"wp-block-heading\" id=\"h-universidade-regional-do-cariri-2009\">Universidade Regional do Cariri (2009)<\/h3><p class=\"wp-block-paragraph\">Sabe&shy;-se que a <span id=\"urn:enhancement-86286fa7-3a57-4ddc-b34e-10dd8614dbe1\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span> de equa&ccedil;&atilde;o x&sup2;+ y&sup2; &minus;4x&minus;6y+11=0 est&aacute; inscrita no quadrado ABCD . Calcule a medida da diagonal desse quadrado<\/p><p class=\"wp-block-paragraph\"><strong>A)<\/strong> 1 u. c.<br><strong>B)<\/strong> 2 u. c.<br><strong>C)<\/strong> 3 u. c.<br><strong>D)<\/strong> 4 u. c.<br><strong>E)<\/strong> 5 u. c.<\/p><details class=\"wp-block-details is-layout-flow wp-block-details-is-layout-flow\"><summary><strong>Resposta<\/strong><\/summary>\n<p class=\"wp-block-paragraph\">Come&ccedil;amos encontrando o raio (r) da <span id=\"urn:enhancement-05c9a241-295b-4fee-8f9a-abcd73f5f556\" class=\"textannotation disambiguated wl-thing\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/circunferencia\">circunfer&ecirc;ncia<\/span> a partir de sua equa&ccedil;&atilde;o:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"235\" height=\"22\" fetchpriority=\"low\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-119.png\" alt=\"\" class=\"wp-image-147475\"><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Agrupe os termos em x e y:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"238\" height=\"26\" fetchpriority=\"low\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-120.png\" alt=\"\" class=\"wp-image-147476\"><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"683\" height=\"161\" fetchpriority=\"low\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-121.png\" alt=\"\" class=\"wp-image-147477\"><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"560\" height=\"52\" fetchpriority=\"low\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-122.png\" alt=\"\" class=\"wp-image-147478\"><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><br><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"593\" height=\"152\" fetchpriority=\"low\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-123.png\" alt=\"\" class=\"wp-image-147479\"><\/figure>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" width=\"529\" height=\"174\" fetchpriority=\"low\" src=\"https:\/\/vestibulares.estrategia.com\/portal\/wp-content\/uploads\/2025\/10\/image-124.png\" alt=\"\" class=\"wp-image-147480\"><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Alternativa correta: <strong>D<\/strong><\/p>\n<\/details><p class=\"wp-block-paragraph\"><\/p><p class=\"wp-block-paragraph\"><\/p><h2 class=\"wp-block-heading\" id=\"h-consiga-a-aprovacao-com-o-estrategia-vestibulares\"><span class=\"ez-toc-section\" id=\"Consiga-a-aprovacao-com-o-Estrategia-Vestibulares\"><\/span>Consiga a aprova&ccedil;&atilde;o com o Estrat&eacute;gia Vestibulares!<span class=\"ez-toc-section-end\"><\/span><\/h2><p class=\"wp-block-paragraph\">Alcance sua vaga t&atilde;o esperada na universidade com o Estrat&eacute;gia Vestibulares! Aqui voc&ecirc; tem acesso a materiais completos e atualizados. Foque nos conte&uacute;dos que mais caem nas provas com nossos planos personalizados, direcionando tempo e <span id=\"urn:enhancement-c2c969e7-a6b6-4fcf-92c4-6ea7bb783906\" class=\"textannotation disambiguated wl-creative-work\" itemid=\"https:\/\/data.wordlift.io\/wl110249\/entity\/energia\">energia<\/span> para os assuntos que realmente v&atilde;o fazer diferen&ccedil;a no seu resultado. 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