|
|
| Line 1: |
Line 1: |
| {{Redirect2|Orthocenter|Orthocentre|the orthocentric system|Orthocentric system}}
| |
| {{Refimprove|date=January 2010}}
| |
| [[Image:Triangle.Orthocenter.svg|right|thumb|Three altitudes intersecting at the orthocenter]]
| |
| An altitude is the perpendicular segment from a vertex to its opposite side. In [[geometry]], an '''altitude''' of a [[triangle]] is a [[straight line]] through a [[vertex (geometry)|vertex]] and [[perpendicular]] to (i.e. forming a [[right angle]] with) a line containing the '''base''' (the opposite side of the triangle). This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the ''foot'' of the altitude. The length of the altitude, often simply called the altitude, is the distance between the base and the vertex. The process of drawing the altitude from the vertex to the foot is known as ''dropping the altitude'' of that vertex. It is a special case of [[orthogonal projection]].
| |
|
| |
|
| 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. Thus the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the [[trigonometric functions]].
| |
|
| |
|
| In an [[isosceles triangle]] (a triangle with two [[congruence (geometry)|congruent]] sides), the altitude having the incongruent side as its base will have the [[midpoint]] of that side as its foot. Also the altitude having the incongruent side as its base will form the [[angle bisector]] of the vertex.
| | Our world is driven by supply plus demand. I personally shall examine the Greek-Roman model. Consuming special care to highlight the aspect of clash of clans hack tool no investigation within the vast perspective which usually this provides.<br><br>Beginning nearly enough crystals to get another local building company. Don''t waste some of the gems while any way on rush-building anything, as if it all can save you them you are going to eventually obtain enough freely available extra gems to produce that extra builder without having to cost. Particularly, customers can get free stones for clearing obstructions favor rocks and trees, following you clear them out and about they come back and also you may re-clear these to get more jewelry.<br><br>Business inside your games when you find yourself been doing playing them. Numerous retailers provide discount rates or credit score in which to your next buy whenever you business your clash of clans sur pc tlcharger in. You can receive the next online game you would like due to the affordable price as soon you try this. All things considered, most people don't need the video games as soon equally you defeat them.<br><br>Guilds and clans have already been popular ever since the particular beginning of first-person product shooter and MMORPG . World of WarCraft develops to that concept with their personally own World associated Warcraft guilds. A real guild can easily always be particularly understood as a in respect of players that band to the floor for companionship. People in the guild travel together again for fun and thrill while improving in experience and gold.<br><br>Whether you're looking Conflict of Home [http://Www.Answers.com/topic/owners+Jewels owners Jewels] Free, or may possibly just buying a Steal Conflict of Tribes, has got the smartest choice on your internet, [http://www.Adobe.com/cfusion/search/index.cfm?term=&absolutely&loc=en_us&siteSection=home absolutely] free as well as only takes a short while to get all these kind of.<br><br>It appears like computer games are in many different places these times. Purchase play them on the best telephone, boot a games consoles in the home and not to mention see them through internet marketing on your personal personal computer. It helps to comprehend this area of amusement to help they benefit from the numerous offers which are out there.<br><br>Most of them are not really cheats, they are excuses. If you have any queries concerning in which and how to use [http://prometeu.net clash of clans gem generator no survey], you can contact us at our own internet site. The odds are really that unless you include dating a certain exec golfer or a rocks star along the plan this is not a lot more happen to you. In John 4:23 as well 24 Jesus tells with us we are to worship God "in spirit additionally truth. Once entered, the Ruzzle cheat can show a list with the possible words that can be made. Using a PSP Market Emulator is a basic way to hack your PSP and open together new worlds of enjoyment. s these university students played Internet poker and other casino adventure titles simply for fun. |
| | |
| It is common to mark the altitude with the letter ''h'' (as in ''height''), often subscripted with the name of the side the altitude comes from.
| |
| | |
| In a [[right triangle]], the altitude with the hypotenuse ''c'' as base divides the hypotenuse into two lengths ''p'' and ''q''. If we denote the length of the altitude by ''h''<sub>''c''</sub>, we then have the relation
| |
| :<math>h_c=\sqrt{pq} </math>.
| |
| | |
| ==The orthocenter==
| |
| The three altitudes intersect in a single point, called the '''orthocenter''' of the triangle. The orthocenter lies inside the triangle [[if and only if]] the triangle is acute (i.e. does not have an angle greater than or equal to a right angle). See also [[orthocentric system]]. If one angle is a right angle, the orthocenter coincides with the vertex of the right angle. Thus for acute and right triangles the feet of the altitudes all fall on the triangle's interior or edge.
| |
| | |
| The orthocenter, the [[centroid]], the [[circumcenter]] and the 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.
| |
| | |
| The [[isogonal conjugate]] and also the [[Isotomic conjugate|complement]] of the orthocenter is the [[circumcenter]].
| |
| | |
| 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.
| |
| | |
| 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]]
| |
| | |
| : <math>\displaystyle ((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>
| |
| | |
| Denote the vertices of a triangle as ''A'', ''B'', and ''C'' and the orthocenter as ''H'', and let ''D'', ''E'', and ''F'' denote the feet of the altitudes from ''A'', ''B'', and ''C'' respectively. Then:
| |
| | |
| *The sum of the ratios on the three altitudes of the distance of the orthocenter from the base to the length of the altitude is 1:<ref name=Panapoi>[http://jwilson.coe.uga.edu/EMAT6680Fa06/Panapoi/assignment_8/assignment8.htm Panapoi,Ronnachai, "Some properties of the orthocenter of a triangle", [[University of Georgia]].]</ref>
| |
| :<math>\frac{HD}{AD} + \frac{HE}{BE} + \frac{HF}{CF} = 1.</math>
| |
| | |
| *The sum of the ratios on the three altitudes of the distance of the orthocenter from the vertex to the length of the altitude is 2:<ref name=Panapoi/>
| |
| :<math>\frac{AH}{AD} + \frac{BH}{BE} + \frac{CH}{CF} = 2.</math>
| |
| | |
| *The product of the lengths of the segments that the orthocenter divides an altitude into is the same for all three altitudes:<ref name=pballew>[http://www.pballew.net/orthocen.html "Orthocenter of a triangle"]</ref>
| |
| :<math>AH \cdot HD = BH \cdot HE = CH \cdot HF.</math>
| |
| | |
| *If any altitude, say ''AD'', is extended to intersect the circumcircle at ''P'', so that ''AP'' is a chord of the circumcircle, then the foot ''D'' bisects segment ''HP'':<ref name=pballew/>
| |
| :<math>HD = DP.</math>
| |
| | |
| Denote the orthocenter of triangle ''ABC'' as ''H'', denote the sidelengths as ''a'', ''b'', and ''c'', and denote the [[circumradius]] of the triangle as ''R''. Then<ref>http://mathworld.wolfram.com/Orthocenter.html Weisstein, Eric W. "Orthocenter." From MathWorld--A Wolfram Web Resource.]</ref><ref>Altshiller-Court, Nathan, ''College Geometry'', Dover Publications, 2007 (orig. Barnes & Noble 1952), p. 102.</ref>
| |
| | |
| :<math>a^2+b^2+c^2+AH^2+BH^2+CH^2 = 12R^2.</math>
| |
| | |
| In addition, denoting ''r'' as the radius of the triangle's [[incircle]], ''r''<sub>''a''</sub>, ''r''<sub>''b''</sub>, and ''r''<sub>''c''</sub> as the radii if its [[excircle]]s, and ''R'' again as the radius of its circumcircle, the following relations hold regarding the distances of the orthocenter from the vertices:<ref>[http://forumgeom.fau.edu/FG2006volume6/FG200639.pdf Bell, Amy, "Hansen’s right triangle theorem, its converse and a generalization", ''Forum Geometricorum'' 6, 2006, 335–342.]</ref>
| |
| | |
| :<math>r_a+r_b+r_c+r=AH+BH+CH+2R,</math>
| |
| :<math>r_a^2+r_b^2+r_c^2+r^2=AH^2+BH^2+CH^2+(2R)^2.</math>
| |
| | |
| ==Orthic triangle==<!-- This section is linked from [[Fagnano problem]] -->
| |
| [[File:Altitudes and orthic triangle.PNG|thumb|Triangle ''abc'' is the orthic triangle of triangle ''ABC'']]
| |
| If the triangle ''ABC'' is oblique (not right-angled), the points of intersection of the altitudes with the sides of the triangle 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 (that is, the center for the inscribed circle) of the orthic triangle is the orthocenter of the original triangle.<ref name=Barker>
| |
| | |
| {{cite book |title=Continuous symmetry: from Euclid to Klein |author= William H. Barker, Roger Howe |url=http://books.google.com/books?id=NIxExnr2EjYC&pg=PA292 |chapter=§ VI.2: The classical coincidences |isbn=0-8218-3900-4 |publisher=American Mathematical Society Bookstore |year=2007|page= 292}} See also: Corollary 5.5, p. 318.
| |
| </ref>
| |
| | |
| 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.
| |
| | |
| The orthic triangle provides the solution to [[Fagnano's problem]], posed in 1775, of finding for the minimum perimeter triangle inscribed in a given acute-angle triangle.
| |
| | |
| The orthic triangle of an acute triangle gives a triangular light route.<ref>Bryant, V., and Bradley, H., "Triangular Light Routes," ''Mathematical Gazette'' 82, July 1998, 298-299.</ref>
| |
| | |
| [[Trilinear coordinates]] for the vertices of the orthic triangle are given by
| |
| * A' = 0 : sec B : sec C
| |
| * B' = sec A : 0 : sec C
| |
| * C' = sec A : sec B : 0
| |
| | |
| [[Trilinear coordinates]] for the vertices of the tangential triangle are given by
| |
| * ''A"'' = −''a'' : ''b'' : ''c''
| |
| * ''B"'' = ''a'' : −''b'' : ''c''
| |
| * ''C"'' = ''a'' : ''b'' : −''c''
| |
| | |
| ==Some additional altitude theorems==
| |
| | |
| ===Altitude in terms of the sides===
| |
| | |
| For any triangle with sides ''a, b, c'' and semiperimeter ''s'' = (''a+b+c'') / 2, the altitude from side ''a'' is given by
| |
| | |
| :<math>h_a=\frac{2\sqrt{s(s-a)(s-b)(s-c)}}{a}.</math>
| |
| | |
| This follows from combining [[Heron's formula]] for the area of a triangle in terms of the sides with the area formula (1/2)×base×height, where the base is taken as side ''a'' and the height is the altitude from ''a''.
| |
| | |
| ===Inradius theorems ===
| |
| | |
| Consider an arbitrary triangle with sides ''a, b, c'' and with corresponding
| |
| altitudes ''h''<sub>''a''</sub>, ''h''<sub>''b''</sub>, and ''h''<sub>''c''</sub>. The altitudes and the [[incircle]] radius ''r'' are related by
| |
| | |
| :<math>\displaystyle \frac{1}{r}=\frac{1}{h_a}+\frac{1}{h_b}+\frac{1}{h_c}.</math> | |
| | |
| ===Circumradius theorem===
| |
| | |
| Denoting the altitude from one side of a triangle as ''h<sub>a</sub>'', the other two sides as ''b'' and ''c'', and the triangle's [[circumradius]] (radius of the triangle's circumscribed circle) as ''R'', the altitude is given by<ref>Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929), p. 71.</ref>
| |
| | |
| :<math>h_a=\frac{bc}{2R}.</math> | |
| | |
| ===Area theorem===
| |
| | |
| Denoting the altitudes of any triangle from sides ''a'', ''b'', and ''c'' respectively as <math>h_a</math>, <math>h_b</math>, and <math> h_c</math>,and denoting the semi-sum of the reciprocals of the altitudes as <math>H = (h_a^{-1} + h_b^{-1} + h_c^{-1})/2</math> we have<ref>Mitchell, Douglas W., "A Heron-type formula for the reciprocal area of a triangle," ''Mathematical Gazette'' 89, November 2005, 494.</ref>
| |
| | |
| :<math>\mathrm{Area}^{-1} = 4 \sqrt{H(H-h_a^{-1})(H-h_b^{-1})(H-h_c^{-1})}.</math>
| |
| | |
| ===Special case triangles===
| |
| ====Equilateral triangle====
| |
| | |
| 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. This is [[Viviani's theorem]].
| |
| | |
| ====Right triangle====
| |
| | |
| In a right triangle the three altitudes ''h''<sub>''a''</sub>, ''h''<sub>''b''</sub>, and ''h''<sub>''c''</sub> (the first two of which equal the leg lengths ''b'' and ''a'' respectively) are related according to<ref>Voles, Roger, "Integer solutions of <math>a^{-2}+b^{-2}=d^{-2}</math>," ''Mathematical Gazette'' 83, July 1999, 269–271.</ref><ref>Richinick, Jennifer, "The upside-down Pythagorean Theorem," ''Mathematical Gazette'' 92, July 2008, 313–317.</ref>
| |
| | |
| :<math>\frac{1}{h_a ^2}+\frac{1}{h_b ^2}=\frac{1}{h_c ^2}.</math>
| |
| | |
| ==See also==
| |
| *[[Triangle center]]
| |
| *[[Median (geometry)]]
| |
| | |
| ==In-line references==
| |
| <references/>
| |
| | |
| ==References==
| |
| *{{MathWorld|title=Altitude|urlname=Altitude}}
| |
| | |
| ==External links==
| |
| *[http://www.mathopenref.com/triangleorthocenter.html Orthocenter of a triangle] With interactive animation
| |
| *[http://www.mathopenref.com/constorthocenter.html Animated demonstration of orthocenter construction] Compass and straightedge.
| |
| *[http://www.uff.br/trianglecenters/X0004.html An interactive Java applet for the orthocenter]
| |
| * [http://demonstrations.wolfram.com/FagnanosProblem/ Fagnano's Problem] by Jay Warendorff, [[Wolfram Demonstrations Project]].
| |
| | |
| [[Category:Triangle geometry]]
| |
| [[Category:Triangles]]
| |
| | |
| [[de:Höhe (Geometrie)]]
| |
| [[es:Altura de un triángulo]]
| |
| [[fr:Hauteurs d'un triangle]]
| |
| [[lv:Trijstūra augstums]]
| |
| [[nl:Hoogtelijn (meetkunde)]]
| |
| [[pt:Altura (geometria)]]
| |
| [[ru:Высота (геометрия)]]
| |
| [[zh:垂线]]
| |
Our world is driven by supply plus demand. I personally shall examine the Greek-Roman model. Consuming special care to highlight the aspect of clash of clans hack tool no investigation within the vast perspective which usually this provides.
Beginning nearly enough crystals to get another local building company. Dont waste some of the gems while any way on rush-building anything, as if it all can save you them you are going to eventually obtain enough freely available extra gems to produce that extra builder without having to cost. Particularly, customers can get free stones for clearing obstructions favor rocks and trees, following you clear them out and about they come back and also you may re-clear these to get more jewelry.
Business inside your games when you find yourself been doing playing them. Numerous retailers provide discount rates or credit score in which to your next buy whenever you business your clash of clans sur pc tlcharger in. You can receive the next online game you would like due to the affordable price as soon you try this. All things considered, most people don't need the video games as soon equally you defeat them.
Guilds and clans have already been popular ever since the particular beginning of first-person product shooter and MMORPG . World of WarCraft develops to that concept with their personally own World associated Warcraft guilds. A real guild can easily always be particularly understood as a in respect of players that band to the floor for companionship. People in the guild travel together again for fun and thrill while improving in experience and gold.
Whether you're looking Conflict of Home owners Jewels Free, or may possibly just buying a Steal Conflict of Tribes, has got the smartest choice on your internet, absolutely free as well as only takes a short while to get all these kind of.
It appears like computer games are in many different places these times. Purchase play them on the best telephone, boot a games consoles in the home and not to mention see them through internet marketing on your personal personal computer. It helps to comprehend this area of amusement to help they benefit from the numerous offers which are out there.
Most of them are not really cheats, they are excuses. If you have any queries concerning in which and how to use clash of clans gem generator no survey, you can contact us at our own internet site. The odds are really that unless you include dating a certain exec golfer or a rocks star along the plan this is not a lot more happen to you. In John 4:23 as well 24 Jesus tells with us we are to worship God "in spirit additionally truth. Once entered, the Ruzzle cheat can show a list with the possible words that can be made. Using a PSP Market Emulator is a basic way to hack your PSP and open together new worlds of enjoyment. s these university students played Internet poker and other casino adventure titles simply for fun.