{"id":4012,"date":"2021-08-13T18:30:03","date_gmt":"2021-08-13T18:30:03","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4012"},"modified":"2021-11-30T16:27:36","modified_gmt":"2021-11-30T10:57:36","slug":"orthocenter-in-a-triangle","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/orthocenter-in-a-triangle\/","title":{"rendered":"Orthocenter in a Triangle | Equation of Locus"},"content":{"rendered":"

Here you will learn formula for orthocenter in a triangle, ex-centers and equation of locus of a point.<\/p>\n

Let’s begin –<\/p>\n

Orthocenter in a Triangle<\/h2>\n

It is the point of intersection of perpendiculars drawn from the vertices on the opposite sides of a triangle and it can be obtained by solving the equation of any two altitudes.<\/p>\n

Note :<\/strong><\/p>\n

\n

(i)\u00a0 If the triangle is right angled, the orthocenter is the point where right angle is formed.<\/p>\n

(ii)\u00a0 Co-ordinates of circumcenter is (\\(x_1tanA+x_2tanB+x_3tanC\\over {tanA+tanB+tanC}\\),\\(y_1tanA+y_2tanB+y_3tanC\\over {tanA+tanB+tanC}\\))<\/p>\n<\/blockquote>\n

Ex-centers<\/h2>\n

The center o a circle which touches side BC and the extended portions of sides AB and AC is called the ex-center of triangle ABC with respect to the vertex A. It is denoted by \\(I_1\\) and its coordinates are<\/p>\n

\n

\\(I_1\\) (\\(-ax_1+bx_2+cx_3\\over {-a+b+c}\\),\\(-ay_1+by_2+cy_3\\over {-a+b+c}\\))<\/p>\n<\/blockquote>\n

Similarly ex-centers of triangle ABC with respect to vertices B and C are denoted by \\(I_2\\) and \\(I_3\\) respectively, and<\/p>\n

\n

\\(I_2\\) (\\(ax_1-bx_2+cx_3\\over {a-b+c}\\),\\(ay_1-by_2+cy_3\\over {a-b+c}\\)), \\(I_3\\) (\\(ax_1+bx_2-cx_3\\over {a+b-c}\\),\\(ay_1+by_2-cy_3\\over {a+b-c}\\))<\/p>\n<\/blockquote>\n

Locus – Equation of locus<\/h2>\n

The locus of a moving point is the path traced out by that point under one or more geometrical conditions<\/p>\n

(a) Equation of Locus :<\/strong><\/p>\n

The equation to a locus is the relation which exists between the coordinates of any point on the path, and which holds for no other point except those lying on the path.<\/p>\n

(b) Procedure for finding the equation of the locus of a point :<\/strong><\/p>\n

(i)\u00a0 If we are finding the equation of the locus of a point P, assign coordinates (h,k) to P.<\/p>\n

(ii)\u00a0 Express the given condition as equations in terms of the known quantities to facilitate calculations. We sometimes include some unknown quantities known as parameters.<\/p>\n

(iii)\u00a0 Eliminate the parameters, so that the eliminant contains only h,k and known quantities.<\/p>\n

(iv) Replace h by x, and k by y, in the eliminant. The resulting equation would be the equation of the locus of P.<\/p>\n\n\n

\n
Next – Condition of Concurrency of Lines<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Circumcenter of a Triangle | Incenter of Triangle<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn formula for orthocenter in a triangle, ex-centers and equation of locus of a point. Let’s begin – Orthocenter in a Triangle It is the point of intersection of perpendiculars drawn from the vertices on the opposite sides of a triangle and it can be obtained by solving the equation of any …<\/p>\n

Orthocenter in a Triangle | Equation of Locus<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[32],"tags":[640,638,639],"yoast_head":"\nOrthocenter in a Triangle | Equation of Locus - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn formula for orthocenter in a triangle and how to find equation of locus of a point.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/orthocenter-in-a-triangle\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Orthocenter in a Triangle | Equation of Locus - 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