Altitude (triangle): Difference between revisions
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{{Redirect2|Orthocenter|Orthocentre|the orthocentric system|Orthocentric system}} |
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[[Image:Triangle.Orthocenter.svg|right|thumb|Orthocenter]] |
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In [[geometry]], an '''altitude''' of a [[triangle (geometry)|triangle]] is a [[straight line]] through a [[vertex (geometry)|vertex]] and [[perpendicular]] to (i.e. forming a [[right angle]] with) the opposite side or an extension of the opposite side. The intersection between the (extended) side and the altitude is called the ''foot'' of the altitude. This opposite side is called the ''base'' of the altitude. The length of the altitude is the distance between the base and the vertex. |
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In an [[isosceles triangle]] (a triangle with two [[congruent]] sides), the altitude having the incongruent side as its base will have the [[midpoint]] of that side as its foot. |
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Altitudes can be used to compute the [[area]] of a triangle: one half of the product of an altitude's length and its base's length equals the triangle's area. Through trigonometric functions, it can also give the length of one side of the triangle.{{clarifyme|date=March 2009}} |
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In a [[right triangle]], the altitude with the hypotenuse as base divides the hypotenuse into two lengths ''p'' and ''q''. If we denote the length of the altitude by ''h'', we then have the relation |
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:''h''<sup>2</sup> = ''pq''. |
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==The orthocenter== |
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The three altitudes intersect in a single point, called the '''orthocenter''' of the triangle. The orthocenter lies inside the triangle (and consequently the feet of the altitudes all fall on the triangle) [[if and only if]] the triangle is not obtuse (i.e. does not have an angle greater than a right angle). See also [[orthocentric system]]. |
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The orthocenter, along with the [[centroid]], [[circumcenter]] and center of the [[nine-point circle]] all lie on a single line, known as the [[Euler line]]. The center of the nine-point circle lies at the [[midpoint]] between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. |
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The [[isogonal conjugate]] and also the [[Complement (mathematics)|complement]] of the orthocenter is the [[circumcenter]]. |
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Four points in the plane such that one of them is the orthocenter of the triangle formed by the other three are called an [[orthocentric system]] or orthocentric quadrangle. |
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Let ''A'', ''B'', ''C'' denote the angles of the reference triangle, and let ''a'' = |''BC''|, ''b'' = |''CA''|, ''c'' = |''AB''| be the sidelengths. The orthocenter has [[trilinear coordinates]] sec ''A'' : sec ''B'' : sec ''C'' and [[Barycentric coordinates (mathematics)|barycentric coordinates]] |
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: <math>((a^2+b^2-c^2)(a^2-b^2+c^2) : (a^2+b^2-c^2)(-a^2+b^2+c^2) : (a^2-b^2+c^2)(-a^2+b^2+c^2)).</math> |
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==Orthic triangle==<!-- This section is linked from [[Fagnano problem]] --> |
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If the triangle ''ABC'' is oblique (not right-angled), the points of intersection of the altitudes with the sides of the triangles form another triangle, A'B'C', called the '''orthic triangle''' or '''altitude triangle'''. It is the [[pedal triangle]] of the orthocenter of the original triangle. Also, the incenter of the orthic triangle is the orthocenter of the original triangle. |
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The orthic triangle is closely related to the '''tangential triangle''', constructed as follows: let ''L''<sub>''A''</sub> be the line tangent to the circumcircle of triangle ''ABC'' at vertex ''A'', and define ''L''<sub>''B''</sub> and ''L''<sub>''C''</sub> analogously. Let ''A"'' = ''L''<sub>''B''</sub> ∩ ''L''<sub>''C''</sub>, ''B"'' = ''L''<sub>''C''</sub> ∩ ''L''<sub>''A''</sub>, ''C"'' = ''L''<sub>''C''</sub> ∩ ''L''<sub>''A''</sub>. The tangential triangle, ''A"B"C"'', is [[homothetic]] to the orthic triangle. |
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The orthic triangle provides the solution to [[Giulio Carlo de' Toschi di Fagnano|Fagnano]]'s problem which in 1775 asked for the minimum perimeter triangle inscribed in a given acute-angle triangle. |
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[[Trilinear coordinates]] for the vertices of the orthic triangle are given by |
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* A' = 0 : sec B : sec C |
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* B' = sec A : 0 : sec C |
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* C' = sec A : sec B : 0 |
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[[Trilinear coordinates]] for the vertices of the tangential triangle are given by |
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* ''A"'' = −''a'' : ''b'' : ''c'' |
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* ''B"'' = ''a'' : −''b'' : ''c'' |
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* ''C"'' = ''a'' : ''b'' : −''c'' |
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==Some additional altitude theorems== |
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===Equilateral triangle theorem=== |
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For any point P within an [[equilateral triangle]], the sum of the perpendiculars to the three sides is equal to the altitude of the triangle. |
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===Inradius theorems=== |
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Consider an arbitrary triangle with sides '''a, b, c''' and with corresponding |
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altitudes '''α, β, η'''. The altitudes and [[incircle]] radius '''r''' are related by |
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:::<math>\tfrac{1}{r}=\tfrac{1}{\alpha}+\tfrac{1}{\beta}+\tfrac{1}{\eta}.</math> |
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Let '''c, h, s''' be the sides of 3 squares associated with the right |
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triangle; the square on the [[hypotenuse]], and the triangle's 2 inscribed |
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squares respectively. The sides of these squares '''(c>h>s)''' |
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and the incircle radius '''r''' are related by a similar formula: |
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:::<math>\tfrac{1}{r}=-{\tfrac{1}{c}}+\tfrac{1}{h}+\tfrac{1}{s}.</math> |
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===The symphonic theorem<sup>*</sup>=== |
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In the case of the right triangle, the sides of the 3 squares '''c, h, s''' are |
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related to each other by the ''symphonic theorem'', as are the 3 altitudes '''α, β, η'''. The ''symphonic theorem'' states that triples '''(c<sup>2</sup>,h<sup>2</sup>,s<sup>2</sup>)''' and '''(α<sup>2</sup>,β<sup>2</sup>,η<sup>2</sup>) '''are ''harmonic'', and that triples <math>(\tfrac{1}{c},\tfrac{1}{h},\tfrac{1}{s})</math> and <math>(\tfrac{1}{\alpha},\tfrac{1}{\beta},\tfrac{1}{\eta})</math> are ''Pythagorean'': |
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:::<math>\tfrac{1}{c^2}+\tfrac{1}{h^2}=\tfrac{1}{s^2}\quad ,\quad \tfrac{1}{\alpha ^2}+\tfrac{1}{\beta ^2}=\tfrac{1}{\eta ^2}.</math> |
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==See also== |
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*[[Triangle center]] |
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*[[Median (geometry)]] |
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==External links== |
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*H. Lee Price and Frank R. Bernhart, Pythagorean triples and a new Pythagorean theorem, arxiv.org:math/0701554, (2007) [http://arxiv.org/abs/math/0701554],[*symphonic theorem] |
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*[http://www.mathopenref.com/triangleorthocenter.html Orthocenter of a triangle] With interactive animation |
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*[http://www.mathopenref.com/constorthocenter.html Animated demonstration of orthocenter construction] Compass and straightedge. |
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*[http://www.uff.br/trianglecenters/X0004.html An interactive Java applet for the orthocenter] |
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* [http://demonstrations.wolfram.com/FagnanosProblem/ Fagnano's Problem] by Jay Warendorff, [[Wolfram Demonstrations Project]]. |
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[[Category:Triangles]] |
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[[Category:Triangle geometry]] |
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[[ar:ارتفاع (مثلث)]] |
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[[bg:Височина (триъгълник)]] |
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[[de:Höhenschnittpunkt]] |
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[[el:Μεσοκάθετη ευθύγραμμου τμήματος]] |
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[[es:Ortocentro]] |
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[[fr:Hauteurs d'un triangle]] |
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[[it:Ortocentro]] |
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[[he:גובה (גאומטריה)]] |
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[[ka:სიმაღლე (გეომეტრია)]] |
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[[hu:Háromszög magassága]] |
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[[nl:Hoogtepunt (meetkunde)]] |
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[[km:កំពស់ត្រីកោណ]] |
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[[pl:Wysokość trójkąta]] |
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[[pt:Altura (geometria)]] |
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[[ru:Ортоцентр]] |
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[[sl:višina trikotnika]] |
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[[sr:Висина троугла]] |
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[[vi:Đường cao (tam giác)]] |
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[[zh:垂线]] |
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