Editing Nine-point circle (section)

==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>