Kosnita's theorem

{{short description|Concurrency of lines connecting to certain circles associated with an arbitrary triangle}}

In Euclidean geometry, Kosnita's theorem is a property of certain circles associated with an arbitrary triangle.

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

Their point of concurrence is known as the triangle's Kosnita point (named by Rigby in 1997). It is the isogonal conjugate of the nine-point center. It is triangle center $X(54)$ in Clark Kimberling's list. This theorem is a special case of Dao's theorem on six circumcenters associated with a cyclic hexagon in.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|....}}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|....}}[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.

References

Ion Pătraşcu (2010), [http://recreatiimatematice.ro/arhiva/processed/22010/14_22010_RM22010.pdf A generalization of Kosnita's theorem] (in Romanian)

John Rigby (1997), Brief notes on some forgotten geometrical theorems. Mathematics and Informatics Quarterly, volume 7, pages 156-158 (as cited by Kimberling).

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

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

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

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]. Forum Geometricorum, volume 14, pages 261–264. {{ISSN|1534-1178}}.

[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}}

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

Category:Theorems about triangles and circles