{"id":2946,"date":"2023-03-25T10:00:07","date_gmt":"2023-03-25T04:30:07","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=2946"},"modified":"2023-03-25T11:05:29","modified_gmt":"2023-03-25T05:35:29","slug":"properties-of-cyclic-quadrilaterals","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/properties-of-cyclic-quadrilaterals\/","title":{"rendered":"What are the Properties of Cyclic Quadrilaterals?"},"content":{"rendered":"<h2>What are the Properties of Cyclic Quadrilaterals?<\/h2>\n<h3>Cyclic quadrilateral<\/h3>\n<p>If all four points of a <a href=\"https:\/\/www.aplustopper.com\/what-is-a-quadrilateral\/\">quadrilateral<\/a> are on circle then it is called cyclic Quadrilateral.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130531 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-1.jpg\" alt=\"What are the Properties of Cyclic Quadrilaterals 1\" width=\"250\" height=\"269\" \/>A quadrilateral PQRS is said to be <strong>cyclic quadrilateral<\/strong> if there exists a circle passing through all its four vertices P, Q, R and S.<br \/>\nLet a cyclic quadrilateral be such that<br \/>\nPQ = a, QR = b, RS = c and SP = d.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130533 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-2.jpg\" alt=\"What are the Properties of Cyclic Quadrilaterals 2\" width=\"213\" height=\"208\" \/>Then \u2220Q + \u2220S = 180\u00b0, \u2220A + \u2220C = 180\u00b0<br \/>\nLet\u00a0 2s = a + b + c + d<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130534 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-3.jpg\" alt=\"What are the Properties of Cyclic Quadrilaterals 3\" width=\"584\" height=\"179\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-3.jpg 584w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-3-300x92.jpg 300w\" sizes=\"(max-width: 584px) 100vw, 584px\" \/><br \/>\n<strong>(1) Circumradius of cyclic quadrilateral:<\/strong> Circum circle of quadrilateral PQRS is also the circumcircle of \u2206PQR.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130535 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-4.jpg\" alt=\"What are the Properties of Cyclic Quadrilaterals 4\" width=\"627\" height=\"73\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-4.jpg 627w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-4-300x35.jpg 300w\" sizes=\"(max-width: 627px) 100vw, 627px\" \/><br \/>\n<strong>(2) Ptolemy&#8217;s theorem:<\/strong> In a cyclic quadrilateral PQRS, the product of diagonals is equal to the sum of the products of the length of the opposite sides i.e., According to Ptolemy&#8217;s theorem, for a cyclic quadrilateral PQRS.<br \/>\nPR.QS = PQ.RS + RQ.PS.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130536 aligncenter\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-5.jpg\" alt=\"What are the Properties of Cyclic Quadrilaterals 5\" width=\"187\" height=\"207\" \/><\/p>\n<h3>Properties of Cyclic Quadrilaterals<\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130537\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-6.jpg\" alt=\"What are the Properties of Cyclic Quadrilaterals 6\" width=\"635\" height=\"377\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-6.jpg 635w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-6-300x178.jpg 300w\" sizes=\"(max-width: 635px) 100vw, 635px\" \/><\/p>\n<p><strong>Theorem: Sum of opposite angles is 180\u00ba (or opposite angles of cyclic quadrilateral is supplementary)<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130538\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-7.png\" alt=\"What are the Properties of Cyclic Quadrilaterals 7\" width=\"253\" height=\"288\" \/><br \/>\nGiven : O is the centre of circle. ABCD is the cyclic quadrilateral.<br \/>\nTo prove : \u2220BAD + \u2220BCD = 180\u00b0, \u2220ABC + \u2220ADC = 180\u00b0<br \/>\nConstruction : Join OB and OD<br \/>\nProof:<br \/>\n(i) \u2220BAD = (\\(\\frac { 1 }{ 2 } \\))\u2220BOD.<br \/>\n(The angle substended by an arc at the centre is double the angle on the circle.)<br \/>\n(ii) \u2220BCD = (\\(\\frac { 1 }{ 2 } \\)) reflex \u2220BOD.<br \/>\n(iii) \u2220BAD + \u2220BCD = (\\(\\frac { 1 }{ 2 } \\))\u2220BOD + (\\(\\frac { 1 }{ 2 } \\)) reflex \u2220BOD.<br \/>\nAdd (i) and (ii).<br \/>\n\u2220BAD + \u2220BCD = (\\(\\frac { 1 }{ 2 } \\))(\u2220BOD + reflex \u2220BOD)<br \/>\n\u2220BAD + \u2220BCD = (\\(\\frac { 1 }{ 2 } \\)) \u00d7 (360\u00b0)<br \/>\n(Complete angle at the centre is 360\u00b0)<br \/>\n\u2220BAD + \u2220BCD = 180\u00b0<br \/>\n(iv) Similarly \u2220ABC + \u2220ADC = 180\u00b0.<\/p>\n<p><iframe loading=\"lazy\" title=\"Opposite Angles of a Cyclic Quadrilateral add up to 180 Degrees - Proof | Don&#039;t Memorise\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/WScvWa_q1j4?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/p>\n<p><strong>Exterior<\/strong> <strong>angle: <\/strong>Exterior angle of cyclic quadrilateral \u00a0is equal to opposite interior angle.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130540\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/What-are-the-Properties-of-Cyclic-Quadrilaterals-8.png\" alt=\"What are the Properties of Cyclic Quadrilaterals 8\" width=\"486\" height=\"338\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/What-are-the-Properties-of-Cyclic-Quadrilaterals-8.png 486w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/What-are-the-Properties-of-Cyclic-Quadrilaterals-8-300x209.png 300w\" sizes=\"(max-width: 486px) 100vw, 486px\" \/><\/p>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/N8tKjFiemF0?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Read More:\u00a0<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/www.aplustopper.com\/different-types-of-quadrilaterals\/\">Different Kinds of Quadrilateral<\/a><\/li>\n<li><a href=\"https:\/\/www.aplustopper.com\/types-of-quadrilaterals\/\">More Solved examples on\u00a0Quadrilaterals<\/a><\/li>\n<li><a href=\"https:\/\/www.aplustopper.com\/lesson\/rs-aggarwal-class-9-solutions-quadrilaterals-and-parallelograms\/\">RS Aggarwal Class 9 Solutions Quadrilaterals and Parallelograms<\/a><\/li>\n<\/ul>\n<h2>Properties of Cyclic Quadrilaterals Example Problems With Solutions<\/h2>\n<p><strong>Example 1: \u00a0 \u00a0<\/strong>Prove that the quadrilateral formed by the internal angle bisectors of any quadrilateral is cyclic.<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130604\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/What-are-the-Properties-of-Cyclic-Quadrilaterals-9.png\" alt=\"What are the Properties of Cyclic Quadrilaterals 9\" width=\"191\" height=\"206\" \/><br \/>\nGiven a quadrilateral ABCD with internal angle bisectors AF, BH, CH and DF of angles A, B, C and D respectively and the points E, F, G and H form a quadrilateral EFGH.<br \/>\nTo prove that EFGH is a cyclic quadrilateral.<br \/>\n\u2220HEF = \u2220AEB [Vertically opposite angles] &#8212;&#8212;&#8211; (1)<br \/>\nConsider triangle AEB,<br \/>\n\u2220AEB + \\(\\frac { 1 }{ 2 } \\) \u2220A + \\(\\frac { 1 }{ 2 } \\) \u2220 B = 180\u00b0<br \/>\n\u2220AEB = 180\u00b0 \u2013 \\(\\frac { 1 }{ 2 } \\) (\u2220A + \u2220 B) &#8212;&#8212;&#8211; (2)<br \/>\nFrom (1) and (2),<br \/>\n\u2220HEF = 180\u00b0 \u2013 \\(\\frac { 1 }{ 2 } \\) (\u2220A + \u2220 B) &#8212;&#8212;&#8212; (3)<br \/>\nSimilarly, \u2220HGF = 180\u00b0 \u2013 \\(\\frac { 1 }{ 2 } \\) (\u2220C + \u2220 D) &#8212;&#8212;&#8211; (4)<br \/>\nFrom 3 and 4,<br \/>\n\u2220HEF + \u2220HGF = 360\u00b0 \u2013 \\(\\frac { 1 }{ 2 } \\) (\u2220A + \u2220B + \u2220C + \u2220 D)<br \/>\n= 360\u00b0 \u2013 \\(\\frac { 1 }{ 2 } \\) (360\u00b0)<br \/>\n= 360\u00b0 \u2013 180\u00b0<br \/>\n= 180\u00b0<br \/>\nSo, EFGH is a cyclic quadrilateral since the sum of the opposite angles of the quadrilateral is 180\u00b0<\/p>\n<p><strong>Example 2: \u00a0 \u00a0<\/strong>ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If \u2220DBC = 70\u00ba, \u2220BAC is 30\u00ba, Find \u2220BCD. Further, if AB = BC, find \u2220ECD.<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130617\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/What-are-the-Properties-of-Cyclic-Quadrilaterals-10.png\" alt=\"What are the Properties of Cyclic Quadrilaterals 10\" width=\"260\" height=\"260\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/What-are-the-Properties-of-Cyclic-Quadrilaterals-10.png 260w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/What-are-the-Properties-of-Cyclic-Quadrilaterals-10-150x150.png 150w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/What-are-the-Properties-of-Cyclic-Quadrilaterals-10-100x100.png 100w\" sizes=\"(max-width: 260px) 100vw, 260px\" \/><br \/>\nFor chord CD,<br \/>\n\u2220CBD = \u2220CAD &#8230; Angles in same segment<br \/>\n\u2220CAD = 70\u00b0<br \/>\n\u2220BAD = \u2220BAC + \u2220CAD = 30\u00b0 + 70\u00b0 = 100\u00b0<br \/>\n\u2220BCD + \u2220BAD = 180\u00b0 &#8230;Opposite angles of a cyclic quadrilateral<br \/>\n\u21d2 \u2220BCD + 100\u00b0 = 180\u00b0<br \/>\n\u21d2 \u2220BCD = 80\u00b0<br \/>\nIn \u25b3ABC<br \/>\nAB = BC (given)<br \/>\n\u2220BCA = \u2220CAB &#8230; Angles opposite to equal sides of a triangle<br \/>\n\u2220BCA = 30\u00b0<br \/>\nAlso, \u2220BCD = 80\u00b0<br \/>\n\u2220BCA + \u2220ACD = 80\u00b0<br \/>\n\u21d2 30\u00b0 + \u2220ACD = 80\u00b0<br \/>\n\u2220ACD = 50\u00b0<br \/>\n\u2220ECD = 50\u00b0<\/p>\n<p><strong>Example 3: \u00a0 \u00a0<\/strong>Prove that a cyclic parallelogram is a rectangle.<br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-130619\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/What-are-the-Properties-of-Cyclic-Quadrilaterals-11.png\" alt=\"What are the Properties of Cyclic Quadrilaterals 11\" width=\"250\" height=\"250\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/What-are-the-Properties-of-Cyclic-Quadrilaterals-11.png 250w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/What-are-the-Properties-of-Cyclic-Quadrilaterals-11-150x150.png 150w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2020\/12\/What-are-the-Properties-of-Cyclic-Quadrilaterals-11-100x100.png 100w\" sizes=\"(max-width: 250px) 100vw, 250px\" \/><br \/>\nGiven that, ABCD is a cyclic parallelogram.<br \/>\nTo prove, ABCD is a rectangle.<br \/>\nProof:<br \/>\n\u22201 + \u22202 = 180\u00b0 &#8230;Opposite angles of a cyclic parallelogram<br \/>\nAlso, Opposite angles of a cyclic parallelogram are equal.<br \/>\nThus,<br \/>\n\u22201 = \u22202<br \/>\n\u21d2 \u22201 + \u22201 = 180\u00b0<br \/>\n\u21d2 \u22201 = 90\u00b0<br \/>\nOne of the interior angle of the parallelogram is right angled. Thus, ABCD is a rectangle.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What are the Properties of Cyclic Quadrilaterals? Cyclic quadrilateral If all four points of a quadrilateral are on circle then it is called cyclic Quadrilateral. A quadrilateral PQRS is said to be cyclic quadrilateral if there exists a circle passing through all its four vertices P, Q, R and S. Let a cyclic quadrilateral be [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[248,1277,1280,1278,1279],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What are the Properties of Cyclic Quadrilaterals? - A Plus Topper<\/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:\/\/www.aplustopper.com\/properties-of-cyclic-quadrilaterals\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What are the Properties of Cyclic Quadrilaterals?\" \/>\n<meta property=\"og:description\" content=\"What are the Properties of Cyclic Quadrilaterals? Cyclic quadrilateral If all four points of a quadrilateral are on circle then it is called cyclic Quadrilateral. A quadrilateral PQRS is said to be cyclic quadrilateral if there exists a circle passing through all its four vertices P, Q, R and S. 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Cyclic quadrilateral If all four points of a quadrilateral are on circle then it is called cyclic Quadrilateral. A quadrilateral PQRS is said to be cyclic quadrilateral if there exists a circle passing through all its four vertices P, Q, R and S. Let a cyclic quadrilateral be [&hellip;]","og_url":"https:\/\/www.aplustopper.com\/properties-of-cyclic-quadrilaterals\/","og_site_name":"A Plus Topper","article_publisher":"https:\/\/www.facebook.com\/aplustopper\/","article_published_time":"2023-03-25T04:30:07+00:00","article_modified_time":"2023-03-25T05:35:29+00:00","og_image":[{"url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-1.jpg"}],"twitter_card":"summary","twitter_misc":{"Written by":"Veerendra","Est. reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/www.aplustopper.com\/#organization","name":"Aplus Topper","url":"https:\/\/www.aplustopper.com\/","sameAs":["https:\/\/www.facebook.com\/aplustopper\/"],"logo":{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/#logo","inLanguage":"en-US","url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg","contentUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg","width":1585,"height":375,"caption":"Aplus Topper"},"image":{"@id":"https:\/\/www.aplustopper.com\/#logo"}},{"@type":"WebSite","@id":"https:\/\/www.aplustopper.com\/#website","url":"https:\/\/www.aplustopper.com\/","name":"A Plus Topper","description":"Improve your Grades","publisher":{"@id":"https:\/\/www.aplustopper.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.aplustopper.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/properties-of-cyclic-quadrilaterals\/#primaryimage","inLanguage":"en-US","url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-1.jpg","contentUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/04\/What-are-the-Properties-of-Cyclic-Quadrilaterals-1.jpg","width":250,"height":269,"caption":"What are the Properties of Cyclic Quadrilaterals 1"},{"@type":"WebPage","@id":"https:\/\/www.aplustopper.com\/properties-of-cyclic-quadrilaterals\/#webpage","url":"https:\/\/www.aplustopper.com\/properties-of-cyclic-quadrilaterals\/","name":"What are the Properties of Cyclic Quadrilaterals? 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