View source for Kosnita's theorem

{{short description|Concurrency of lines connecting to certain circles associated with an arbitrary triangle}}
[[File:Kosnita points.svg|300px|right|thumb|X(54) is Kosnita point of the triangle ABC]]

In [[Euclidean geometry]], '''Kosnita's theorem''' is a property of certain [[Circle|circle]]s associated with an arbitrary [[Triangle|triangle]].

Let <math>ABC</math> be an arbitrary triangle, <math>O</math> its circumcenter and <math>O_a,O_b,O_c</math> are the circumcenters of three triangles <math>OBC</math>, <math>OCA</math>, and <math>OAB</math> respectively.  The theorem claims that the three straight lines <math>AO_a</math>, <math>BO_b</math>, and <math>CO_c</math> are concurrent.<ref name=wolframKosnita/> This result was established by the Romanian mathematician Cezar Coşniţă (1910-1962).<ref name=patrascu/>

Their point of concurrence is known as the triangle's '''Kosnita point''' (named by Rigby in 1997).  It is the [[Isogonal conjugate|isogonal conjugate]] of the nine-point center.<ref name=grinberg2003/><ref name=rigby1997/> It is [[Triangle center|triangle center]] <math>X(54)</math> in Clark Kimberling's list.<ref name=kimberlingX54/> This theorem is special case of Dao's theorem on six circumcenters associated with a cyclic hexagon in.<ref name=dergiadesDao6CCCH/><ref name=cohlDao6CCCH/><ref name=NgoQuangDuong /><ref name=C.Kimberling/><ref>Nguyễn Minh Hà, ''[http://geometry-math-journal.ro/pdf/Volume6-Issue1/4.pdf Another Purely Synthetic Proof of Dao's Theorem on Sixcircumcenters]''.  Journal of Advanced Research on Classical and Modern Geometries,  {{ISSN|2284-5569}}, volume 6, pages 37–44. {{MR|....}}</ref><ref>Nguyễn Tiến Dũng, ''[http://geometry-math-journal.ro/pdf/Volume6-Issue1/6.pdf A Simple proof of Dao's Theorem on Sixcircumcenters]''.  Journal of Advanced Research on Classical and Modern Geometries, {{ISSN|2284-5569}}, volume 6, pages 58–61. {{MR|....}}</ref><ref>[http://www.journal-1.eu/2016-3/Nguyen-Ngoc-Giang-The-extension-pp.21-32.pdf The extension from a circle to a conic having center: The creative method of new theorems], International Journal of Computer Discovered Mathematics, pp.21-32.</ref>

== References ==
<references>
<ref name=patrascu>Ion Pătraşcu (2010), ''[http://recreatiimatematice.ro/arhiva/processed/22010/14_22010_RM22010.pdf A generalization of Kosnita's theorem]'' (in Romanian)</ref>

<ref name=rigby1997>John Rigby (1997), ''Brief notes on some forgotten geometrical theorems.'' Mathematics and Informatics Quarterly, volume 7, pages 156-158 (as cited by Kimberling).</ref>

<ref name=grinberg2003>Darij Grinberg (2003), ''[http://forumgeom.fau.edu/FG2003volume3/FG200311.pdf On the Kosnita Point and the Reflection Triangle].'' [[Organization:Forum Geometricorum|Forum Geometricorum]], volume 3, pages 105–111. {{ISSN|1534-1178}}</ref>

<ref name=kimberlingX54>Clark Kimberling (2014), ''[http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X54 Encyclopedia of Triangle Centers] {{webarchive|url=https://web.archive.org/web/20120419171900/http://faculty.evansville.edu/ck6/encyclopedia/ETC.html |date=2012-04-19 }}'', section ''X(54) = Kosnita Point''.  Accessed on 2014-10-08</ref>

<ref name=dergiadesDao6CCCH>Nikolaos Dergiades (2014), ''[http://forumgeom.fau.edu/FG2014volume14/FG201424.pdf Dao's Theorem on Six Circumcenters associated with a Cyclic Hexagon].'' [[Organization:Forum Geometricorum|Forum Geometricorum]], volume 14, pages=243–246. {{ISSN|1534-1178}}.</ref>

<ref name=cohlDao6CCCH>Telv Cohl (2014), ''[http://forumgeom.fau.edu/FG2014volume14/FG201429index.html A purely synthetic proof of Dao's theorem on six circumcenters associated with a cyclic hexagon].'' [[Organization:Forum Geometricorum|Forum Geometricorum]], volume 14, pages 261–264. {{ISSN|1534-1178}}.</ref>

<ref name=NgoQuangDuong>[http://www.journal-1.eu/2016-2/Ngo-Quang-Duong-Dao-theorem-pp.40-47.pdf Ngo Quang Duong, International Journal of Computer Discovered Mathematics, Some problems around the Dao's theorem on six circumcenters associated with a cyclic hexagon configuration], volume 1, pages=25-39. {{ISSN|2367-7775}}</ref>

<ref name=C.Kimberling>Clark Kimberling (2014), [http://faculty.evansville.edu/ck6/encyclopedia/ETCPart3.html#X3649 X(3649) = KS(INTOUCH TRIANGLE)]</ref>

<ref name=wolframKosnita>{{mathworld|id=KosnitaTheorem|title=Kosnita Theorem}}</ref>
</references>

[[Category:Triangle geometry]]
[[Category:Theorems in plane geometry]]

{{Sourceattribution|Kosnita's theorem|1}}