View source for Nine-point hyperbola

{{short description|Hyperbola constructed from a given triangle and point}} 
[[File:Nine point conic.svg|right|thumb|400px|
{{legend|blue|Points of triangle {{math|△''ABC''}} and given point {{mvar|P}}}}
{{legend-line|solid #a17123|Six constituent lines of the [[complete quadrangle|quadrangle]] formed by {{mvar|A, B, C, P}}}}
{{legend-line|solid #188f61|'''Nine-point hyperbola'''. The right branch [[Bisection|bisects]] {{mvar|{{overline|BA}}, {{overline|BC}}, {{overline|BP}}}}; the left bisects {{mvar|{{overline|PA}}, {{overline|PC}}, {{overline|AC}}}}, and passes through the [[Line–line intersection|intersections]] of lines {{mvar|BC, PA}} and {{mvar|AB, PC}}.}}]]

In [[Euclidean geometry]] with [[triangle]] {{math|△''ABC''}}, the '''nine-point hyperbola''' is an instance of the [[nine-point conic]] described by American mathematician [[Maxime Bôcher]] in 1892. The celebrated [[nine-point circle]] is a separate instance of Bôcher's conic:
:Given a triangle {{math|△''ABC''}} and a point {{mvar|P}} in its plane, a [[Conic section|conic]] can be drawn through the following nine points:
:: the [[midpoint]]s of the sides of {{math|△''ABC''}},
:: the midpoints of the lines joining {{mvar|P}} to the vertices, and
:: the points where these last named lines cut the sides of the triangle.
The conic is an [[ellipse]] if {{mvar|P}} lies in the interior of {{math|△''ABC''}} or in one of the regions of the plane separated from the interior by two sides of the triangle; otherwise, the conic is a [[hyperbola]]. Bôcher notes that when {{mvar|P}} is the [[orthocenter]], one obtains the nine-point circle, and when {{mvar|P}} is on the [[circumcircle]] of {{math|△''ABC''}}, then the conic is an equilateral hyperbola.

==Allen==

An approach to the nine-point hyperbola using the [[analytic geometry]] of [[split-complex number]]s was devised by E. F. Allen in 1941.<ref>Allen, E.F. (1941) "On a Triangle Inscribed in a Rectangular Hyperbola", [[American Mathematical Monthly]] 48, No.10 pp.&nbsp;675–681</ref> Writing <math>z=a+bj</math>, {{math|1=''j''{{sup|2}} = 1}}, he uses split-complex arithmetic to express a hyperbola as
:<math>z z^* = a^2 .</math>
It is used as the [[circumconic]] of triangle <math>t_1, t_2, t_3 .</math> Let <math>s = t_1 + t_2 + t_3 .</math> Then the nine-point conic is
:<math>\left(z - \frac{s}{2}\right)\left( z^* - \frac{s^*}{2}\right) = \frac{a^2}{4} .</math>
Allen's description of the nine-point hyperbola followed a development of the [[nine-point circle]] that [[Frank Morley]] and his son published in 1933. They requisitioned the [[unit circle]] in the [[complex plane]] as the [[circumcircle]] of the given triangle.

In 1953 Allen extended his study to a nine-point conic of a triangle inscribed in any central conic.<ref>E. F. Allen (1953) "An extended inversive geometry", ''American Mathematical Monthly'' 60(4):233–7</ref>

==Yaglom==
For Yaglom, a hyperbola is a ''Minkowskian circle'' as in the [[Minkowski plane]]. Yaglom's description of this geometry is found in the "Conclusion" chapter of a book that initially addresses Galilean geometry.<ref>[[Isaak Yaglom]] (1979) ''A Simple Non-Euclidean Geometry and its Physical Basis'', page 193</ref> He considers a triangle inscribed in a "circumcircle" which is in fact a hyperbola. In the Minkowski plane the nine-point hyperbola is also described as a circle:
:… the midpoints of the sides of a triangle {{math|△''ABC''}} and the feet of its altitudes (as well as the midpoints of the segments joining the orthocenter of {{math|△''ABC''}} to its vertices) lie on a [Minkowskian] circle {{mvar|S}} whose radius is half the radius of the circumcircle of the triangle. It is natural to refer to S as the six- (nine-) point circle of the (Minkowskian) triangle {{math|△''ABC''}}; if {{math|△''ABC''}} has an [[Incircle and excircles of a triangle|incircle]] {{mvar|s}}, then the six- (nine-) point circle {{mvar|S}} of {{math|△''ABC''}} touches its incircle {{mvar|s}} (Fig.173).

==Others==
In 2005 J. A. Scott<ref>J. A. Scott (2005) "A Nine-point Hyperbola", [[The Mathematical Gazette]] 89:93–6 (#514)</ref> used the [[unit hyperbola]] as the [[circumconic]] of triangle ABC and found conditions for it to include six triangle centers: the [[centroid]] X(2), the [[orthocenter]] X(4), the [[Fermat point]]s X(13) and X(14), and the [[Napoleon points]] X(17) and X(18) as listed in the [[Encyclopedia of Triangle Centers]]. Scott’s hyperbola is a [[Kiepert conics|Kiepert hyperbola]] of the triangle.

Christopher Bath<ref>Christopher Bath (2010) [http://people.bath.ac.uk/masgcs/Article95.pdf A Nine Point Rectangular Hyperbola]</ref> describes a nine-point rectangular hyperbola passing through these centers: [[incenter]] X(1), the three [[excenter]]s, the centroid X(2), the [[de Longchamps point]] X(20), and the three points obtained by extending the triangle [[median (geometry)|median]]s to twice their [[cevian]] length.

==References==
{{Reflist}}

* [[Maxime Bôcher]] (1892) [https://www.jstor.org/stable/1967142 Nine-point Conic], [[Annals of Mathematics]], link from [[Jstor]].
* Maud A. Minthorn (1912) [http://babel.hathitrust.org/cgi/pt?id=uc1.b3808276;view=1up;seq=1 The Nine Point Conic], Master's dissertation at [[University of California, Berkeley]], link from [[HathiTrust]]. Maude Ellen Minthorn, born 1883, LeMars, Iowa, died 1966, St. Petersburg, Florida. Dau. of Pennington Minthorn, 1856-1939, (brother of Hulda Minthorn Hoover, mother of Pres. Herbert Hoover) and Anna Mary Heald, 1887-1940, (sister of Franklin Herman Heald, founder of Lake Elsinore, California (WRH) Maud Minthorn also taught at Fresno, Cal, High School 1935-1940 
* Bjørn Felsager (2004) [https://web.archive.org/web/20090106144140/http://www.dynamicgeometry.com/documents/advancedSketchGallery/minkowski/Minkowski_Workshop_1.pdf Minkowski geometry, Part 1], [https://web.archive.org/web/20081121024206/http://www.dynamicgeometry.com/documents/advancedSketchGallery/minkowski/Minkowski_Workshop_2.pdf Minkowski geometry, Part 2] ICME-10 Copenhagen.

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[[Category:Curves defined for a triangle]]