{"id":155456,"date":"2022-03-26T00:39:58","date_gmt":"2022-03-25T19:09:58","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/cevas-theorem\/"},"modified":"2023-01-02T12:54:38","modified_gmt":"2023-01-02T07:24:38","slug":"cevas-theorem","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/","title":{"rendered":"Ceva\u2019s Theorem"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#Introduction_to_Cevas_Theorem\" title=\"Introduction to Ceva\u2019s Theorem\">Introduction to Ceva\u2019s Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#Statement_of_Cevas_Theorem\" title=\"Statement of Ceva\u2019s Theorem\">Statement of Ceva\u2019s Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#Cevas_Theorem_Proof\" title=\"Ceva\u2019s Theorem Proof\">Ceva\u2019s Theorem Proof<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#Converse_of_Cevas_Theorem\" title=\"Converse of Ceva\u2019s Theorem\">Converse of Ceva\u2019s Theorem<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#First_Approach\" title=\"First Approach\">First Approach<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#Second_Approach\" title=\"Second Approach\">Second Approach<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Introduction_to_Cevas_Theorem\"><\/span>Introduction to Ceva\u2019s Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Ceva\u2019s theorem is a theorem in geometry that states that the sum of the interior angles of a triangle is equal to 180 degrees. The theorem is named after the Italian mathematician Giovanni Ceva, who published it in 1678.<\/p>\n<p>The theorem can be proven using basic geometric reasoning. Consider a triangle ABC. The sum of the interior angles of a triangle is 180 degrees. Therefore, the angle at A is 180 degrees &#8211; the sum of the angles at B and C. Similarly, the angle at B is 180 degrees &#8211; the sum of the angles at A and C, and the angle at C is 180 degrees &#8211; the sum of the angles at A and B. This proves that the sum of the interior angles of a triangle is 180 degrees.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-155455 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/cevas-theorem.jpg\" alt=\"\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/cevas-theorem.jpg?v=1648235394 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/cevas-theorem-300x212.jpg?v=1648235394 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Statement_of_Cevas_Theorem\"><\/span>Statement of Ceva\u2019s Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Ceva\u2019s theorem states that the sum of the cevians of a triangle is zero. A cevian is a line segment from a vertex of a triangle to the midpoint of the opposite side.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Cevas_Theorem_Proof\"><\/span>Ceva\u2019s Theorem Proof<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Proof of Ceva\u2019s theorem<\/p>\n<p>Let ABC be a triangle with points D, E, and F on its sides.<\/p>\n<p>We will show that the line segments AD, BE, and CF are concurrent.<\/p>\n<p>To do so, we will use the following theorem:<\/p>\n<p>Theorem: If two lines intersect in a point, then the lines are perpendicular.<\/p>\n<p>Proof:<\/p>\n<p>Given: Two lines intersect in a point.<\/p>\n<p>Prove: The lines are perpendicular.<\/p>\n<p>Given: AB and CD intersect in point O.<\/p>\n<p>Prove: AB is perpendicular to CD.<\/p>\n<p>Since AB and CD intersect in a point, they must be parallel.<\/p>\n<p>Since they are parallel, by the parallel postulate, they must be the same distance apart.<\/p>\n<p>Since they are the same distance apart, by the triangle inequality, the length of AB is less than the length of CD.<\/p>\n<p>Since the length of AB is less than the length of CD, by the definition of perpendicularity, AB is perpendicular to CD.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Converse_of_Cevas_Theorem\"><\/span>Converse of Ceva\u2019s Theorem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>If three points are on a line, then the line is the perpendicular bisector of the segment connecting the points.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"First_Approach\"><\/span>First Approach<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The first approach is to use the built-in truncate() function. The truncate() function truncates a string to the specified number of characters.<\/p>\n<p>The following example truncates the string &#8220;Hello, world!&#8221; to &#8220;Hello&#8221; .<\/p>\n<p>string truncate(string input, int length)<\/p>\n<p>input &#8211; The string to truncate.<br \/>\nlength &#8211; The number of characters to truncate.<\/p>\n<p>The following example truncates the string &#8220;Hello, world!&#8221; to &#8220;Hello&#8221; .<\/p>\n<p>string truncate(&#8220;Hello, world!&#8221;, 5)<\/p>\n<p>Hello<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Second_Approach\"><\/span>Second Approach<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The second approach to solving this problem is to use the fact that the sum of two consecutive positive integers is always two more than the larger of the two numbers. We can use this fact to find the sum of the first two positive integers, and then use that sum to find the sum of the next two positive integers.<\/p>\n<p>The sum of the first two positive integers is 1 + 2 = 3. The sum of the next two positive integers is 3 + 4 = 7. Therefore, the sum of the first five positive integers is 3 + 7 + 10 + 13 + 16 = 59.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Introduction to Ceva\u2019s Theorem Ceva\u2019s theorem is a theorem in geometry that states that the sum of the interior angles [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Ceva\u2019s Theorem","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn about Ceva\u2019s Theorem topic of Maths in details explained by subject experts on infinitylearn.com. Register free for online.","custom_permalink":"maths\/cevas-theorem\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Ceva\u2019s Theorem - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"Learn about Ceva\u2019s Theorem topic of Maths in details explained by subject experts on infinitylearn.com. Register free for online.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Ceva\u2019s Theorem - Infinity Learn by Sri Chaitanya\" \/>\n<meta property=\"og:description\" content=\"Learn about Ceva\u2019s Theorem topic of Maths in details explained by subject experts on infinitylearn.com. Register free for online.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/\" \/>\n<meta property=\"og:site_name\" content=\"Infinity Learn by Sri Chaitanya\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/\" \/>\n<meta property=\"article:published_time\" content=\"2022-03-25T19:09:58+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-01-02T07:24:38+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/cevas-theorem.jpg\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@InfinityLearn_\" \/>\n<meta name=\"twitter:site\" content=\"@InfinityLearn_\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"vipin\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Ceva\u2019s Theorem - Infinity Learn by Sri Chaitanya","description":"Learn about Ceva\u2019s Theorem topic of Maths in details explained by subject experts on infinitylearn.com. Register free for online.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/","og_locale":"en_US","og_type":"article","og_title":"Ceva\u2019s Theorem - Infinity Learn by Sri Chaitanya","og_description":"Learn about Ceva\u2019s Theorem topic of Maths in details explained by subject experts on infinitylearn.com. Register free for online.","og_url":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/","og_site_name":"Infinity Learn by Sri Chaitanya","article_publisher":"https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","article_published_time":"2022-03-25T19:09:58+00:00","article_modified_time":"2023-01-02T07:24:38+00:00","og_image":[{"url":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/cevas-theorem.jpg"}],"twitter_card":"summary_large_image","twitter_creator":"@InfinityLearn_","twitter_site":"@InfinityLearn_","twitter_misc":{"Written by":"vipin","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/infinitylearn.com\/surge\/#organization","name":"Infinity Learn","url":"https:\/\/infinitylearn.com\/surge\/","sameAs":["https:\/\/www.facebook.com\/InfinityLearn.SriChaitanya\/","https:\/\/www.instagram.com\/infinitylearn_by_srichaitanya\/","https:\/\/www.linkedin.com\/company\/infinity-learn-by-sri-chaitanya\/","https:\/\/www.youtube.com\/c\/InfinityLearnEdu","https:\/\/twitter.com\/InfinityLearn_"],"logo":{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/#logo","inLanguage":"en-US","url":"","contentUrl":"","caption":"Infinity Learn"},"image":{"@id":"https:\/\/infinitylearn.com\/surge\/#logo"}},{"@type":"WebSite","@id":"https:\/\/infinitylearn.com\/surge\/#website","url":"https:\/\/infinitylearn.com\/surge\/","name":"Infinity Learn by Sri Chaitanya","description":"Surge","publisher":{"@id":"https:\/\/infinitylearn.com\/surge\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/infinitylearn.com\/surge\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#primaryimage","inLanguage":"en-US","url":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/cevas-theorem.jpg?v=1648235394","contentUrl":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/cevas-theorem.jpg?v=1648235394","width":606,"height":428},{"@type":"WebPage","@id":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#webpage","url":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/","name":"Ceva\u2019s Theorem - Infinity Learn by Sri Chaitanya","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/#website"},"primaryImageOfPage":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#primaryimage"},"datePublished":"2022-03-25T19:09:58+00:00","dateModified":"2023-01-02T07:24:38+00:00","description":"Learn about Ceva\u2019s Theorem topic of Maths in details explained by subject experts on infinitylearn.com. Register free for online.","breadcrumb":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/infinitylearn.com\/surge\/"},{"@type":"ListItem","position":2,"name":"Ceva\u2019s Theorem"}]},{"@type":"Article","@id":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#article","isPartOf":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#webpage"},"author":{"@id":"https:\/\/infinitylearn.com\/surge\/#\/schema\/person\/d931698bc4645b2739855720864f30e2"},"headline":"Ceva\u2019s Theorem","datePublished":"2022-03-25T19:09:58+00:00","dateModified":"2023-01-02T07:24:38+00:00","mainEntityOfPage":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#webpage"},"wordCount":515,"publisher":{"@id":"https:\/\/infinitylearn.com\/surge\/#organization"},"image":{"@id":"https:\/\/infinitylearn.com\/surge\/maths\/cevas-theorem\/#primaryimage"},"thumbnailUrl":"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/cevas-theorem.jpg","articleSection":["Maths"],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/infinitylearn.com\/surge\/#\/schema\/person\/d931698bc4645b2739855720864f30e2","name":"vipin","image":{"@type":"ImageObject","@id":"https:\/\/infinitylearn.com\/surge\/#personlogo","inLanguage":"en-US","url":"https:\/\/secure.gravatar.com\/avatar\/c9a84adf9d11e7ad01332089c3e52538?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/c9a84adf9d11e7ad01332089c3e52538?s=96&d=mm&r=g","caption":"vipin"},"sameAs":["http:\/\/surge.infinitylearn.com"],"url":"https:\/\/infinitylearn.com\/surge\/author\/vipin\/"}]}},"_links":{"self":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts\/155456"}],"collection":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/comments?post=155456"}],"version-history":[{"count":0,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/posts\/155456\/revisions"}],"wp:attachment":[{"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/media?parent=155456"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/categories?post=155456"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/tags?post=155456"},{"taxonomy":"table_tags","embeddable":true,"href":"https:\/\/infinitylearn.com\/surge\/wp-json\/wp\/v2\/table_tags?post=155456"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}