Editing Nine-point circle

{{Short description|Circle constructed from a triangle}}
[[File:Triangle.NinePointCircle.svg|200px|thumb|The nine points]]

<!-- [[File:EulerCircle3.gif|thumb|You can change the vertices of the triangle, and Euler's circle persists.]] -->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 [[point (geometry)|points]] are:

* The [[midpoint]] of each side of the triangle
* The [[Perpendicular|foot]] of each [[altitude (triangle)|altitude]]
* 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).<ref>{{harvtxt|Altshiller-Court|1925|pp=103–110}}</ref><ref>{{harvtxt|Kay|1969|pp=18,245}}</ref>

The nine-point circle is also known as  '''Feuerbach's circle''' (after [[Karl Wilhelm Feuerbach]]), '''Euler's circle''' (after [[Leonhard Euler]]),  '''Terquem's circle''' (after [[Olry Terquem]]), the '''six-points circle''', the '''twelve-points circle''', the '''{{mvar|n}}-point circle''', the '''medioscribed circle''', the '''mid circle''' or the '''circum-midcircle'''. Its center is the [[nine-point center]] of the triangle.<ref>{{cite journal|author=Kocik, Jerzy|author2=Solecki, Andrzej|title=Disentangling a Triangle|journal=Amer. Math. Monthly|volume=116|issue=3|year=2009|pages=228–237|url=http://www.maa.org/programs/maa-awards/writing-awards/disentangling-a-triangle|doi=10.4169/193009709x470065}} Kocik and Solecki (sharers of a 2010 [[Lester R. Ford Award]]) give a proof of the Nine-Point Circle Theorem.</ref><ref>{{cite book|author=Casey, John|author-link=John Casey (mathematician)|title=''Nine-Point Circle Theorem, in'' A Sequel to the First Six Books of Euclid|page=58|year=1886|edition=4th|location=London|publisher=Longmans, Green, & Co|url=http://babel.hathitrust.org/cgi/pt?id=hvd.hn6mqv;view=1up;seq=78}}</ref>

==Nine Significant Points of Nine Point Circle==

[[File:Nine-point circle.svg]]

The diagram above shows the nine significant points of the nine-point circle. Points {{mvar|D, E, F}} are the midpoints of the three sides of the triangle. Points {{mvar|G, H, I}} are the feet of the altitudes of the triangle. Points {{mvar|J, K, L}} are the midpoints of the line segments between each altitude's [[vertex (geometry)|vertex]] intersection (points {{mvar|A, B, C}}) and the triangle's orthocenter (point {{mvar|S}}).

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.

==Discovery==

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'' {{mvar|D, E, F, G, H, 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'' {{mvar|J, K, L}}.)  Thus, Terquem was the first to use the name nine-point circle.

==Tangent circles==
[[File:Circ9pnt3.svg|right|thumb|250px|The nine-point circle is tangent to the incircle and excircles.]]
In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally [[Tangent circles|tangent]] to that triangle's three [[excircle]]s and internally tangent to its [[incircle]]; this result is known as [[Feuerbach's theorem]].  He proved that:<blockquote>... 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...</blockquote> {{sfn|Feuerbach|Buzengeiger|1822}}

The [[triangle center]] at which the incircle and the nine-point circle touch is called the [[Feuerbach point]].

==Other properties of the nine-point circle==

* The radius of a triangle's [[circumscribed circle|circumcircle]] is twice the radius of that triangle's nine-point circle.<ref name=PL/>{{rp|p.153}}
[[File:9pcircle03.svg]]
''Figure 3''

* A nine-point circle bisects a line segment going from the corresponding triangle's orthocenter to any point on its circumcircle.
[[File:9pcircle 04.png]]
''Figure 4''

* The center {{mvar|N}} of the nine-point circle bisects a segment from the orthocenter {{mvar|H}} to the [[circumscribed circle|circumcenter]] {{mvar|O}} (making the orthocenter a center of [[homothetic center|dilation]] to both circles):<ref name=PL>Posamentier, Alfred S., and Lehmann, Ingmar. ''[[The Secrets of Triangles]]'', Prometheus Books, 2012.</ref>{{rp|p.152}}

::<math>\overline{ON} = \overline{NH}.</math>

* The nine-point center {{mvar|N}} is one-fourth of the way along the [[Euler line]] from the centroid {{mvar|G}} to the orthocenter {{mvar|H}}:<ref name=PL/>{{rp|p.153}}

::<math>\overline{HN} = 3\overline{NG}.</math>

* Let {{math|ω}}  be the nine-point circle of the diagonal triangle of a [[cyclic quadrilateral]]. The point of intersection of the bimedians of the cyclic quadrilateral belongs to the nine-point circle.<ref>{{Cite journal|last=Fraivert|first=David|date=July 2019|title=New points that belong to the nine-point circle|journal=The Mathematical Gazette|volume=103|issue=557|pages=222–232|doi=10.1017/mag.2019.53|s2cid=213935239 }}</ref><ref>{{Cite journal|last=Fraivert|first=David|date=2018|title=New applications of method of complex numbers in the geometry of cyclic quadrilaterals|url=https://ijgeometry.com/wp-content/uploads/2018/04/5-16.pdf|journal=International Journal of Geometry|volume=7|issue=1|pages=5–16}}</ref>
[[File:Nine-point circle of diagonal triangle.png|thumb|{{mvar|ABCD}} is a cyclic quadrilateral. {{math|△''EFG''}} is the diagonal triangle of {{mvar|ABCD}}. The point {{mvar|T}} of intersection of the bimedians of {{mvar|ABCD}} belongs to the nine-point circle of {{math|△''EFG''}}.]]

* The nine-point circle of a reference triangle is the circumcircle of both the reference triangle's [[medial triangle]] (with vertices at the midpoints of the sides of the reference triangle) and its [[orthic triangle]] (with vertices at the feet of the reference triangle's altitudes).<ref name=PL/>{{rp|p.153}}
* The center of all [[rectangular hyperbola]]s that pass through the vertices of a triangle lies on its nine-point circle. Examples include the well-known rectangular hyperbolas of [[Friedrich Wilhelm August Ludwig Kiepert|Keipert]], [[Václav Jeřábek|Jeřábek]] and Feuerbach. This fact is known as the Feuerbach conic theorem.

[[File:Tangent circles in Feuerbach's theorem.jpg|thumb|The nine point circle and the 16 tangent circles of the orthocentric system]]

* If an [[orthocentric system]] of four points {{mvar|A, B, C, H}} is given, then the four triangles formed by any combination of three distinct 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.)
* Consequently, these four triangles have circumcircles with identical radii. Let {{mvar|N}} represent the common nine-point center and {{mvar|P}} be an arbitrary point in the plane of the orthocentric system. Then

::<math>\overline{NA}^2 + \overline{NB}^2 + \overline{NC}^2 + \overline{NH}^2 = 3R^2</math> 

:where {{mvar|R}} is the common [[circumradius]]; and if

::<math>\overline{PA}^2 + \overline{PB}^2 + \overline{PC}^2 + \overline{PH}^2 = K^2,</math>

:where {{mvar|K}} is kept constant, then the locus of {{mvar|P}} is a circle centered at {{mvar|N}} with a radius <math>\tfrac{1}{2} \sqrt{K^2-3R^2}.</math> As {{mvar|P}} approaches {{mvar|N}} the locus of {{mvar|P}} for the corresponding constant {{mvar|K}}, collapses onto {{mvar|N}} the nine-point center. Furthermore the nine-point circle is the locus of {{mvar|P}} such that

::<math>\overline{PA}^2 + \overline{PB}^2 + \overline{PC}^2 + \overline{PH}^2 = 4R^2.</math>

* 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.
* If four arbitrary points {{math|A, B, C, D}} are given that do not form an orthocentric system, then the nine-point circles of {{math|△''ABC'', △''BCD'', △''CDA'', △''DAB''}} concur at a point, the [[Poncelet point]] of {{mvar|A, B, C, D}}. 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.
* If four points {{math|A, B, C, D}} are given that form a [[cyclic quadrilateral]], then the nine-point circles of {{math|△''ABC'', △''BCD'', △''CDA'', △''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 transformation|homothetic]] to the reference cyclic quadrilateral {{mvar|ABCD}} by a factor of –{{sfrac|1|2}} and its homothetic center {{mvar|N}} lies on the line connecting the circumcenter {{mvar|O}} to the anticenter {{mvar|M}} where

::<math>\overline{ON} = 2\overline{NM}.</math>

* The [[orthopole]] of lines passing through the circumcenter lie on the nine-point circle.
* A triangle's circumcircle, its nine-point circle, its [[polar circle (geometry)|polar circle]], and the circumcircle of its [[tangential triangle]]<ref>{{harvtxt|Altshiller-Court|1925|p=98}}</ref> are [[coaxal circles|coaxal]].<ref>{{harvtxt|Altshiller-Court|1925|p=241}}</ref>
* [[Trilinear coordinates]] for the center of the [[Kiepert hyperbola]] are

::<math>\frac{(b^2 -c^2)^2}{a} : \frac{(c^2-a^2)^2}{b} : \frac{(a^2-b^2)^2}{c}</math>

* Trilinear coordinates for the center of the Jeřábek hyperbola are

::<math>\cos(A)\sin^2(B-C) : \cos(B)\sin^2(C-A) : \cos(C)\sin^2(A-B)</math>

*  Letting {{math|''x'' : ''y'' : ''z''}} be a variable point in trilinear coordinates, an equation for the nine-point circle is

:: <math>x^2\sin 2A + y^2\sin 2B + z^2\sin 2C-2(yz\sin A + zx\sin B + xy\sin C) = 0.</math>

==Generalization==
{{Main|Nine-point conic}}
The circle is an instance of a [[conic section]] and the nine-point circle is an instance of the general nine-point conic that has been constructed with relation to a triangle {{math|△''ABC''}} and a fourth point {{mvar|P}}, where the particular nine-point circle instance arises when {{mvar|P}} is the orthocenter of {{math|△''ABC''}}. The vertices of the triangle and {{mvar|P}} determine a [[complete quadrilateral]] and three "diagonal points" where opposite sides of the quadrilateral intersect. There are six "sidelines" in the quadrilateral; the nine-point conic intersects the midpoints of these and also includes the diagonal points. The conic is an [[ellipse]] when {{mvar|P}} is interior to {{math|△''ABC''}} or in a region sharing [[vertical angles]] with the triangle, but a [[nine-point hyperbola]] occurs when {{mvar|P}} is in one of the three adjacent regions, and the hyperbola is rectangular when P lies on the circumcircle of {{math|△''ABC''}}.

==See also==
* [[Hart circle]], a related construction for [[circular triangle]]s
* [[Lester's theorem]]
* [[Poncelet point]]
* [[Synthetic geometry]]

== Notes ==
{{reflist}}

== References ==

* {{ citation | first1 = Nathan | last1 = Altshiller-Court | year = 1925 | lccn = 52013504 | title = College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle | edition = 2nd | publisher = [[Barnes & Noble]] | location = New York }}
* {{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 }}.
* {{ citation | first1 = David C. | last1 = Kay | year = 1969 | lccn = 69012075 | title = College Geometry | publisher = [[Holt, Rinehart and Winston]] | location = New York }}
* {{ citation | first1 = David | last1 = Fraivert 
 | journal = [[The Mathematical Gazette]]
 | pages = 222–232
 | title = New points that belong to the nine-point circle
 | volume = 103
 | issue = 557
 | year = 2019| doi = 10.1017/mag.2019.53 | s2cid = 213935239 
 }}
* {{ citation | first1 = David | last1 = Fraivert 
 | journal = [[International Journal of Geometry]]
 | pages = 5–16
 | title = New applications of method of complex numbers in the geometry of cyclic quadrilaterals
 | url = https://ijgeometry.com/wp-content/uploads/2018/04/5-16.pdf
 | volume = 7
 | issue = 1
 | year = 2018}}

== External links ==
*[https://www.geogebra.org/m/kzabj78a Nine-point circle] - interactive illustration of the nine-point circle and some of its properties
*[https://rykap.com/nine_point_circle/index.html "A Javascript demonstration of the nine point circle"] at rykap.com
*[https://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).
* 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]
* {{mathworld|urlname=Nine-PointCircle|title=Nine-Point Circle}}
* {{mathworld|urlname=Orthopole|title=Orthopole}}
* [https://www.cut-the-knot.org/Curriculum/Geometry/SixPointCircle.shtml Nine Point Circle] at [[cut-the-knot]]
* [https://demonstrations.wolfram.com/TheCenterAndRadiusOfTheNinePointCircle/ Interactive Nine Point Circle applet] from the Wolfram Demonstrations Project
* [https://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.
* [https://arxiv.org/pdf/0806.3617.pdf N J Wildberger. Chromogeometry.] Discusses the nine-point circle with regard to three different quadratic forms (blue, red, green).

[[Category:Circles defined for a triangle]]