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| [[Image:Triangle.NinePointCircle.svg|200px|thumb|The nine points]]
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| In [[geometry]], the '''nine-point circle''' is a [[circle]] that can be constructed for any given [[triangle]]. It is so named because it passes through nine significant [[concyclic points]] defined from the triangle. These nine points are:
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| * The [[midpoint]] of each side of the [[triangle]]
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| * The foot of each [[altitude (triangle)|altitude]]
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| * The [[midpoint]] of the [[line segment]] from each [[vertex (geometry)|vertex]] of the triangle to the [[orthocenter]] (where the three altitudes meet; these line segments lie on their respective altitudes).
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| The nine-point circle is also known as '''Feuerbach's circle''', '''Euler's circle''', '''Terquem's circle''', the '''six-points circle''', the '''twelve-points circle''', the '''''n''-point circle''', the '''medioscribed circle''', the '''mid circle''' or the '''circum-midcircle'''. | | The next step to this game''s success is that may it produces the impression that it''s a multi-player game. I believe it''s a fantasy due to the you don''t do can be necessary directly with any other player. You don''t fight and explore in unison like you would of Wow, of play in opposition of another player even for with a turn-by-turn basis comparable to Chess. Any time you raid another player''s village, that player is offline combined with you could at you see, the same time just end up raiding a random computer-generated village.<br><br>Go on the internet for help. Just about any game has its possess legion of devoted devotees, lots of which spend countless hours crafting comprehensive maps and guides. Additionally there are newsgroups where you are qualified to speak one on with other players. Benefit from this found diamond and it is possible to eventually get past that much cla you have been into on forever.<br><br>clash of clans is a ideal game, which usually requires one to build your personal village, discover warriors, raid profits and build your extremely clan and so forward. there is a lot a lot somewhat more to this video game and for every such kinds of you require jewels in order to really play, as you which includes. Clash of Clans hack allows you to get as many jewels as you want. There is an unlimited regarding gems you could yield with all the Clash of Clans cheats ready online, however you need to be specific about the link you are using on account of some of them just now waste materials your serious amounts of also dont get anybody anything more.<br><br>Unswervingly check several distinct stores before purchasing a sports. Be sure to look both online combined with in [http://www.adobe.com/cfusion/search/index.cfm?term=&genuine&loc=en_us&siteSection=home genuine] brick and simply mortar stores in your region. The costs of a video video game may differ widely, unusually if a game is not brand new. By performing a modest additional leg work, it's is possible to look for clash of clans.<br><br>No matter the reason, computer game tips are widespread and sent out fairly rapidly over the web. The gaming community is trying to find means steer clear of cheaters from overrunning whichever game; having lots involved with cheaters playing a unattached game can really make honest players to naturally quit playing, or play just with friends they trust. This poses a extremely popular problem particularly for request games for example EverQuest, wherein a loss connected with players ultimately result within a loss of income.<br><br>Rare metal and Elixir would function as main sources available on the inside Clash of Clans. If you adored this article and you also would like to [http://www.Adobe.com/cfusion/search/index.cfm?term=&receive&loc=en_us&siteSection=home receive] more info pertaining to [http://prometeu.net how to hack clash of clans] kindly visit the web-page. Each of these regarding are necessary and could be gathered by a associated with ways. Frontrunners of people can use structures, recover the cash some other tribes perhaps clash of clans hack tools for acquiring them both.<br><br>You don''t necessarily to possess one of the cutting edge troops to win victories. A mass volume of barbarians, your first-level troop, will totally destroy an adversary village, and strangely it''s quite enjoyable to from the virtual carnage. |
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| == Significant nine points ==
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| [[Image:Nine-point circle.svg]]
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| The diagram above shows the nine significant [[point (geometry)|points]] of the nine-point circle. Points ''D'', ''E'', and ''F'' are the [[midpoint]]s of the three sides of the [[triangle]]. Points ''G'', ''H'', and ''I'' are the feet of the [[altitude (triangle)|altitudes]] of the [[triangle]]. Points ''J'', ''K'', and ''L'' are the [[midpoint]]s of the [[line segment]]s between each [[altitude (triangle)|altitude's]] [[vertex (geometry)|vertex]] intersection (points ''A'', ''B'', and ''C'') and the [[triangle|triangle's]] [[orthocenter]] (point ''S'').
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| For an [[acute triangle]], six of the points (the midpoints and altitude feet) lie on the triangle itself; for an [[obtuse triangle]] two of the altitudes have feet outside the triangle, but these feet still belong to the nine-point circle.
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| ==Discovery==
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| Although he is credited for its discovery, [[Karl Wilhelm Feuerbach]] did not entirely discover the nine-point circle, but rather the six point circle, recognizing the significance of the midpoints of the three sides of the triangle and the feet of the altitudes of that triangle. (''See Fig. 1, points'' D, E, F, G, H, ''and'' I.) (At a slightly earlier date, [[Charles Brianchon]] and [[Jean-Victor Poncelet]] had stated and proven the same theorem.) But soon after Feuerbach, mathematician [[Olry Terquem]] himself proved the existence of the circle. He was the first to recognize the added significance of the three midpoints between the triangle's vertices and the [[orthocenter]]. (''See Fig. 1, points'' J, K, ''and'' L.) Thus, Terquem was the first to use the name nine-point circle.
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| ==Tangent circles==
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| [[Image:Circ9pnt3.svg|right|thumb|250px|The nine-point circle is [[tangent]] to the [[incircle]] and [[excircle]]s]]
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| In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally [[tangent]] to that triangle's three [[excircle]]s and internally tangent to its [[incircle]]; this result is known as '''Feuerbach's theorem'''. He postulated that:
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| :''... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle...'' {{harv|Feuerbach|1822}}
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| The point at which the incircle and the nine-point circle touch is often referred to as the '''Feuerbach point'''.
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| ==Other properties of the nine-point circle==
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| * The radius of a triangle's [[circumcircle]] is twice the radius of that triangle's nine-point circle.
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| [[Image:9pcircle03.svg]]
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| ''Figure 3''
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| * A nine-point circle bisects a line segment going from the corresponding triangle's [[orthocenter]] to any [[point (geometry)|point]] on its [[circumcircle]].
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| [[Image:9pcircle 04.png]]
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| ''Figure 4''
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|
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| * The center of any nine-point circle (the '''nine-point center''') lies on the corresponding triangle's [[Euler's line|Euler line]], at the [[midpoint]] between that triangle's [[orthocenter]] and [[circumcenter]].
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| * The nine-point center lies at the [[centroid]] of four points comprising the triangle's three vertices and its orthocenter.
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| * Of the nine points, the three midpoints of line segments between the vertices and the orthocenter are reflections of the triangle's midpoints about its nine-point center.
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| * The center of all rectangular hyperbolas that pass through the vertices of a triangle lies on its nine-point circle. Examples include the well-known rectangular hyperbolas of Keipert, Jeřábek and Feuerbach. This fact is known as the Feuerbach conic theorem.
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| [[File:Tangent circles in Feuerbach's theorem.jpg|thumb|The nine point circle and the 16 tangent circles of the orthocentric system]]
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| * If an [[orthocentric system]] of four [[point (geometry)|points]] ''A'', ''B'', ''C'' and ''H'' is given, then the four [[triangle]]s formed by any combination of three distinct [[point (geometry)|points]] of that system all share the same nine-point circle. This is a consequence of symmetry: the ''sides'' of one triangle adjacent to a vertex that is an orthocenter to another triangle are ''segments'' from that second triangle. A third midpoint lies on their common side. (The same 'midpoints' defining separate nine-point circles, those circles must be concurrent.)
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| * Consequently, these four triangles have circumcircles with identical radii. Let ''N'' represent the common nine-point center and ''P'' be an arbitrary point in the plane of the orthocentric system. Then ''NA''<sup>2</sup>+''NB''<sup>2</sup>+''NC''<sup>2</sup>+''NH''<sup>2</sup> = ''3R''<sup>2</sup> where ''R'' is the common [[circumradius]] and if ''PA''<sup>2</sup>+''PB''<sup>2</sup>+''PC''<sup>2</sup>+''PH''<sup>2</sup> = ''K''<sup>2</sup>, where ''K'' is kept constant, then the locus of ''P'' is a circle centered at ''N'' with a radius <math>\scriptstyle \frac{1}{2} \sqrt{K^2-3R^2}</math>. As ''P'' approaches ''N'' the locus of ''P'' for the corresponding constant ''K'', collapses onto ''N'' the nine-point center. Furthermore the nine-point circle is the locus of ''P'' such that ''PA''<sup>2</sup>+''PB''<sup>2</sup>+''PC''<sup>2</sup>+''PH''<sup>2</sup> = ''4R''<sup>2</sup>.
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| * The centers of the incircle and excircles of a triangle form an orthocentric system. The nine-point circle created for that orthocentric system is the circumcircle of the original triangle. The feet of the altitudes in the orthocentric system are the vertices of the original triangle.
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| * If four arbitrary points ''A'', ''B'', ''C'', ''D'' are given that do not form an orthocentric system, then the nine-point circles of ''ABC'', ''BCD'', ''CDA'' and ''DAB'' concur at a [[point (geometry)|point]]. The remaining six intersection points of these nine-point circles each concur with the midpoints of the four triangles. Remarkably, there exists a unique nine-point conic, centered at the centroid of these four arbitrary points, that passes through all seven points of intersection of these nine-point circles. Furthermore because of the Feuerbach conic theorem mentioned above, there exists a unique rectangular [[circumconic]], centered at the common intersection point of the four nine-point circles, that passes through the four original arbitrary points as well as the orthocenters of the four triangles.
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| * If four points ''A'', ''B'', ''C'', ''D'' are given that form a [[cyclic quadrilateral]], then the nine-point circles of ''ABC'', ''BCD'', ''CDA'' and ''DAB'' concur at the [[Cyclic_quadrilateral#Anticenter_and_collinearities|anticenter]] of the cyclic quadrilateral. The nine-point circles are all congruent with a radius of half that of the cyclic quadrilateral's circumcircle. The nine-point circles form a set of four [[Johnson circles]]. Consequently the four nine-point centers are cyclic and lie on a circle congruent to the four nine-point circles that is centered at the anticenter of the cyclic quadrilateral. Furthermore the cyclic quadrilateral formed from the four nine-pont centers is [[homothetic]] to the reference cyclic quadrilateral ''ABCD'' by a factor of −''<sup>1</sup>/<sub>2</sub>'' and its homothetic center ''(N)'' lies on the line connecting the circumcenter ''(O)'' to the anticenter ''(M)'' where ''ON'' = ''2NM''.
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| * The [[orthopole]] of lines passing through the [[circumcircle|circumcenter]] lie on the nine-point circle.
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| * [[Trilinear coordinates]] for the nine-point center are cos (''B'' − ''C'') : cos (''C'' − ''A'') : cos (''A'' − ''B'')
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| * [[Trilinear coordinates]] for the Feuerbach point are 1 − cos (''B'' − ''C'') : 1 − cos (''C'' − ''A'') : 1 − cos (''A'' − ''B'')
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| * [[Trilinear coordinates]] for the center of the Kiepert hyperbola are (''b''<sup>2</sup>'' − c''<sup>2</sup>)<sup>2</sup>/''a'' : (''c''<sup>2</sup> − ''a''<sup>2</sup>)<sup>2</sup>/''b'' : (''a''<sup>2</sup> − ''b''<sup>2</sup>)<sup>2</sup>/''c''
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| * [[Trilinear coordinates]] for the center of the Jeřábek hyperbola are cos ''A'' sin<sup>2</sup>(''B'' − ''C'') : cos ''B'' sin<sup>2</sup>(''C'' − ''A'') : cos ''C'' sin<sup>2</sup>(''A'' − ''B'')
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| * Letting ''x'' : ''y'' : ''z'' be a variable point in [[trilinear coordinates]], an equation for the nine-point circle is
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| : ''x''<sup>2</sup>sin ''2A'' + ''y''<sup>2</sup>sin 2''B'' + ''z''<sup>2</sup>sin 2''C'' − 2(''y''z sin ''A'' + ''zx'' sin ''B'' + ''xy'' sin ''C'') = 0.
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| ==See also==
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| * [[Lester's theorem]]
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| * [[Nine-point hyperbola]]
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| * [[Poncelet point]]
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| * [[Synthetic geometry]]
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| * [[Triangle center]]
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| == References ==
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| * {{citation | last1 = Feuerbach | first1 = Karl Wilhelm | author1-link = Karl Wilhelm Feuerbach | last2 = Buzengeiger | first2 = Carl Heribert Ignatz | author2-link = Carl Heribert Ignatz Buzengeiger | year = 1822 | title = Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten Linien und Figuren. Eine analytisch-trigonometrische Abhandlung | publisher = Wiessner | location = Nürnberg | edition = Monograph | url = http://resolver.sub.uni-goettingen.de/purl?PPN512512426 }}.
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| == External links ==
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| *[http://rykap.com/ninePointCircle "A Javascript demonstration of the nine point circle"] at rykap.com
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| *[http://faculty.evansville.edu/ck6/encyclopedia/ETC.html ''Encyclopedia of Triangles Centers''] by Clark Kimberling. The nine-point center is indexed as X(5), the Feuerbach point, as X(11), the center of the Kiepert hyperbola as X(115), and the center of the Jeřábek hyperbola as X(125).
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| * History about the nine-point circle based on J.S. MacKay's article from 1892: [http://jwilson.coe.uga.edu/EMT668/EMT668.Folders.F97/Anderson/geometry/geometry1project/historyofninepointcircle/history.html History of the Nine Point Circle]
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| * {{mathworld|urlname=Nine-PointCircle|title=Nine-Point Circle}}
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| * {{mathworld|urlname=Orthopole|title=Orthopole}}
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/SixPointCircle.shtml Nine Point Circle in Java] at [[cut-the-knot]]
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| * [http://www.cut-the-knot.org/Curriculum/Geometry/FeuerbachProof.shtml Feuerbach's Theorem: a Proof] at [[cut-the-knot]]
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| * [http://www.walter-fendt.de/m14e/triangle.htm Special lines and circles in a triangle] by [[Walter Fendt]] (requires Java)
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| * [http://www.uff.br/trianglecenters/nine-point-circle.html An interactive Java applet showing several triangle centers that lies on the Nine Point Circle].
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| * [http://demonstrations.wolfram.com/TheCenterAndRadiusOfTheNinePointCircle/ Interactive Nine Point Circle applet] from the Wolfram Demonstrations Project
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| * [http://dynamicmathematicslearning.com/ninepointconic.html Nine-point conic and Euler line generalization] at [http://dynamicmathematicslearning.com/JavaGSPLinks.htm Dynamic Geometry Sketches] Generalizes nine-point circle to a nine-point conic with an associated generalization of the Euler line.
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| [[Category:Circles]]
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| [[Category:Triangle geometry]]
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