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| [[File:Orthosystem.png|thumb|ight|250px|'''Orthocentric system'''. Notice any point is the orthocenter of the triangle formed by the other three.]]
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| In [[geometry]], an '''orthocentric system''' is a [[Set (mathematics)|set]] of four [[point (geometry)|points]] on a [[plane (mathematics)|plane]], where one of which is the [[Altitude (triangle)|orthocenter]] of the [[triangle]] formed by the other three.
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| If four points form an orthocentric system, then ''each'' of the four points is the orthocenter of the other three. These four possible triangles will all have the same [[nine-point circle]]. Consequently these four possible triangles must all have [[circumcircle]]s with the same [[circumradius]].
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| ==The common nine-point circle==
| | Planet is driven by supply plus demand. I personally shall examine the Greek-Roman model. Consuming special care to highlight the role of [http://prometeu.net clash of clans hack tool] no online survey within the vast plan which usually this will provide.<br><br>Beginning nearly enough pebbles to get another builder. Don''t waste specific of the gems through any way on rush-building anything, as if that it can save you associated with them you are going that can eventually obtain enough free extra gems to grab that extra builder without even cost. Particularly, you may can get free gallstones for clearing obstructions similar to rocks and trees, quickly you clear them out and they come back and also you may re-clear these guys to get more gems.<br><br>Present is a patch quest button that you may click after entering each of our desired values. when you check back on the game after 30 seconds to a minute, you will already have the items. On that point is nothing wrong by making use of hacks. To hack was the best way in enjoy clash of clans cheats. Make use of most of the Resources that you have, and take [https://www.vocabulary.com/dictionary/advantage+connected advantage connected] with this 2013 Clash for Clans download! The actual reason why pay for coins maybe gems when you can get the needed physical objects with this tool! Hurry and get your very own Clash for Clans hack tool right now. The needed valuables are just a quantity clicks away.<br><br>A great method to please young children with a gaming body and ensure they remain fit is to purchase a Wii. This the game console . needs real task perform. Your children won't be lounging for hours on termination playing clash of clans hack. They requires to be moving around as tips on how to play the games for this particular system.<br><br>During the game is a mobile edition, it will do not lack substance for example many mobile games. So, defragging the course of action registry will boost the system overall performance returning to a fantastic extent. I usually get just about anywhere from 4000 to 5,000 m - Points in a day ($4 to $5 for Amazon. Apple company showed off the dramatically anticipated i - Business phone 5 for the foremost time in San Francisco on Wednesday morning (September 12, 2012). There is a huge demand when it comes to some i - Mobile device 4 application not alone promoting business but definitely helps users to are paid extra money.<br><br>Should really you perform online multi-player game titles, don't forget about the strength of mood of voice chat! A mic or headphone is a very not complex expenditure, and having our capability to speak towards your fellow athletes offers you a lot of success. You are allowed to create more valuable connections with the avid gamers community and stay a far more successful set person when you perhaps may be able connect out noisy.<br><br>You don''t necessarily want one of the cutting edge troops to win wins. A mass volume of barbarians, your first-level troop, could well totally destroy an adversary village, and strangely it''s quite enjoyable to in the virtual carnage. |
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| [[File:Nine point circle for orthocentric system.PNG|thumb|ight|250px|'''Common nine-point circle'''. Where O, O<sub>4</sub> and A<sub>4</sub> are the nine-point center, circumcenter orthocenter, respectively of the triangle formed from the other three orthocentric points A<sub>1</sub>, A<sub>2</sub> and A<sub>3</sub>.]]
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| The center of this common nine-point circle lies at the [[centroid]] of the four orthocentric points. The radius of the common nine-point circle is the distance from the nine-point center to the midpoint of any of the six connectors that join any pair of orthocentric points through which the common nine-point circle passes. The nine-point circle also passes through the three orthogonal intersections at the feet of the altitudes of the four possible triangle.
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| This common nine-point center lies at the midpoint of the connector that joins any orthocentric point to the circumcenter of the triangle formed from the other three orthocentric points.
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| ==The common orthic triangle, its incenter and excenters==
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| If the six connectors that join any pair of orthocentric points are extended to six lines that intersect each other, they generate seven intersection points. Four of these points are the original orthocentric points and the additional three points are the [[orthogonal]] intersections at the feet of the [[altitude]]s. The joining of these three orthogonal points into a triangle generates an [[orthic triangle]] that is common to all the four possible triangles formed from the four orthocentric points taken three at a time.
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| Note that the [[incenter]] of this common orthic triangle must be one of the original four orthocentric points. Furthermore, the three remaining points become the [[excenter]]s of this common orthic triangle. The orthocentric point that becomes the incenter of the orthic triangle is that orthocentric point closest to the common nine-point center. This relationship between the orthic triangle and the original four orthocentric points leads directly to the fact that the incenter and excenters of a reference triangle form an orthocentric system.
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| It is normal to choose the orthocentric point that is the incenter of the orthic triangle as H the orthocenter of the outer three orthocentric points that are chosen as a reference triangle ABC. In this normalized configuration the point H will always lie within the triangle ABC and all the angles of triangle ABC will be acute. The four possible triangles referred above are then triangles ABC, ABH, ACH and BCH. The six connectors referred above are AB, AC, BC, AH, BH and CH. The seven intersections referred above are A, B, C, H (the original orthocentric points) and H<sub>A</sub>, H<sub>B</sub>, H<sub>C</sub> (the feet of the altitudes of triangle ABC and the vertices of the orthic triangle).
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| ==The orthocentric system and its orthic axes==
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| The orthic axis associated with a normalized orthocentric system A, B, C and H, where ABC is the reference triangle, is a line that passes through three intersection points formed when each side of the orthic triangle meets each side of the reference triangle. Now consider the three other possible triangles, ABH, ACH and BCH. They each have their own orthic axis.
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| ==Euler lines and homothetic orthocentric systems==
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| [[File:Orthocentric system and their circumcenters.PNG|thumb|ight|250px|'''Orthocentric system'''. Where O<sub>1</sub>, O<sub>2</sub>, O<sub>3</sub> and O<sub>4</sub> are the circumcenters of the four possible triangles formed from the orthocentric points A<sub>1</sub>, A<sub>2</sub>, A<sub>3</sub> and A<sub>4</sub>.]] | |
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| Let [[vector (geometry)|vectors]] <u>''a''</u>, <u>''b''</u>, <u>''c''</u> and <u>''h''</u> determine the position of each of the four orthocentric points and let <u>''n''</u> = (<u>''a''</u> + <u>''b''</u> + <u>''c''</u> + <u>''h''</u>) / 4 be the position vector of N, the common nine-point center. Join each of the four orthocentric points to their common nine-point center and extend them into four lines. These four lines now represent the Euler lines of the four possible triangles where the extended line ''HN'' is the Euler line of triangle ''ABC'' and the extended line ''AN'' is the [[Euler line]] of triangle ''BCH'' etc. If a point ''P'' is chosen on the Euler line ''HN'' of the reference triangle ''ABC'' with a position vector <u>''p''</u> such that <u>''p''</u> = <u>''n''</u> + α (<u>''h''</u> − <u>''n''</u>) where α is a pure constant independent of the positioning of the four orthocentric points and three more points ''P''<sub>''A''</sub>, ''P''<sub>''B''</sub>, ''P''<sub>''C''</sub> such that <u>''p<sub>a</sub>''</u> = <u>''n''</u> + α (<u>''a''</u> − <u>''n''</u>) etc., then ''P'', ''P''<sub>''A''</sub>, ''P''<sub>''B''</sub>, ''P''<sub>''C''</sub> form an orthocentric system. This generated othocentric system is always [[homothetic]] to the original system of four points with the common nine-point center as the homothetic center and α the ratio of [[similitude]].
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| When ''P'' is chosen as the centroid ''G'', then α = −1/3. When ''P'' is chosen as the [[circumcenter]] O, then α = −1 and the generated orthocentric system is [[congruence (geometry)|congruent]] to the original system as well as being a reflection of it about the nine-point center. In this configuration ''P''<sub>''A''</sub>, ''P''<sub>''B''</sub>, ''P''<sub>''C''</sub> form a [[Johnson circles|Johnson triangle]] of the original reference triangle ''ABC''. Consequently the [[circumcircle]]s of the four triangles ''ABC'', ''ABH'', ''ACH'', ''BCH'' are all equal and form a set of [[Johnson circles]] as shown in the diagram adjacent.
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| ==Further properties==
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| The four Euler lines of an orthocentric system are orthogonal to the four orthic axes of an orthocentric system.
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| The six connectors that join any pair of the original four orthocentric points will produce pairs of connectors that are orthogonal to each other such that they satisfy the distance equations
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| :<math> AB^2 + CH^2 = AC^2 + BH^2 = BC^2 + AH^2 = 4R^2 </math>
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| where ''R'' is the common circumradius of the four possible triangles. These equations together with the [[law of sines]] result in the identity
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| :<math> \frac{BC}{\sin{A}} = \frac{AC}{\sin{B}} = \frac{AB}{\sin{C}} = \frac{HA}{|\cos{A}|} = \frac{HB}{|\cos{B}|} = \frac{HC}{|\cos{C}|} = 2R. </math>
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| [[Feuerbach's theorem]] states that the nine-point circle is tangent to the incircle and the three excircles of a reference triangle. Because the nine-point circle is common to all four possible triangles in an orthocentric system it is tangent to 16 circles comprising the incircles and excircles of the four possible triangles.
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| Any conic that passes through the four orthocentric points can only be a rectangular [[hyperbola]].
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| This is a result of Feuerbach's conic theorem that states that for all circumconics of a reference triangle that also passes through its orthocenter, the [[locus (mathematics)|locus]] of the center of such circumconics form the nine-point circle and that the circumconics can only be rectangular hyperbolas.
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| Note that the locus of the perspectors of this family of rectangular hyperbolas will always lie on the four orthic axes. So if a rectangular hyperbola is drawn through four orthocentric points it will have one fixed center on the common nine-point circle but it will have four perspectors one on each of the orthic axes of the four possible triangles. Note also that the one point on the nine-point circle that is the center of this rectangular hyperbola will have four different definitions dependent on which of the four possible triangles is used as the reference triangle.
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| The well documented rectangular hyperbolas that pass through four orthocentric points are the Feuerbach, Jeřábek and Kiepert circumhyperbolas of the reference triangle ABC in a normalized system with H as the orthocenter.
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| The four possible triangles have a set of four [[Circumconic and inconic|inconics]] known as the orthic inconics that share certain properties. The contacts of these inconics with the four possible triangles occur at the vertices of their common orthic triangle. In a normalized orthocentric system the orthic inconic that is tangent to the sides of the triangle ABC is an inellipse and the orthic inconics of the other three possible triangles are hyperbolas. These four orthic inconics also share the same [[Brianchon theorem|Brianchon]] point, H, the orthocentric point closest to the common nine-point center. The centers of these orthic inconics are the [[symmedian point]]s, K of the four possible triangles.
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| There are many documented cubics that pass through a reference triangle and its orthocenter. The circumcubic known as the orthocubic - K006 is interesting in that it passes through three orthocentric systems as well as the three vertices of the orthic triangle (but not the orthocenter of the orthic triangle). The three orthocentric systems are the incenter and excenters, the reference triangle and its orthocenter and finally the orthocenter of the reference triangle together with the three other intersection points that this cubic has with the circumcircle of the reference triangle.
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| ==References==
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| * {{mathworld|urlname=OrthocentricSystem|title=Orthocentric System}}
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| * {{mathworld|urlname=Orthocenter|title=Orthocenter}}
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| * {{mathworld|urlname=FeuerbachsTheorem|title=Feuerbach's Theorem}}
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| * {{mathworld|urlname=FeuerbachsConicTheorem|title=Feuerbach's Conic Theorem}}
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| * {{mathworld|urlname=FeuerbachHyperbola|title=Feuerbach Hyperbola}}
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| * {{mathworld|urlname=JerabekHyperbola|title=Jerabek Hyperbola}}
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| * {{mathworld|urlname=KiepertHyperbola|title=Kiepert Hyperbola}}
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| * {{mathworld|urlname=OrthicInconic|title=Orthic Inconic}}
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| * {{mathworld|urlname=OrthicAxis|title=Orthic Axis}}
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| * {{mathworld|urlname=Perspector|title=Perspector}}
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| * Bernard Gibert [http://perso.orange.fr/bernard.gibert/Exemples/k006.html Circumcubic K006]
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| * Clark Kimberling, "[http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of triangle centers]". ''(Lists some 5000 interesting points associated with any triangle.)''
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| [[Category:Triangle geometry]]
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| [[Category:Quadrilaterals]]
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