Kosnita's theorem
In Euclidean geometry, Kosnita's theorem is a property of certain circles associated with an arbitrary triangle.
Let [math]\displaystyle{ ABC }[/math] be an arbitrary triangle, [math]\displaystyle{ O }[/math] its circumcenter and [math]\displaystyle{ O_a,O_b,O_c }[/math] are the circumcenters of three triangles [math]\displaystyle{ OBC }[/math], [math]\displaystyle{ OCA }[/math], and [math]\displaystyle{ OAB }[/math] respectively. The theorem claims that the three straight lines [math]\displaystyle{ AO_a }[/math], [math]\displaystyle{ BO_b }[/math], and [math]\displaystyle{ CO_c }[/math] are concurrent.[1] This result was established by the Romanian mathematician Cezar Coşniţă (1910-1962).[2]
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.[3][4] It is triangle center [math]\displaystyle{ X(54) }[/math] in Clark Kimberling's list.[5] This theorem is special case of Dao's theorem on six circumcenters associated with a cyclic hexagon in.[6][7][8][9][10][11][12]
References
- ↑ Weisstein, Eric W.. "Kosnita Theorem". http://mathworld.wolfram.com/KosnitaTheorem.html.
- ↑ Ion Pătraşcu (2010), A generalization of Kosnita's theorem (in Romanian)
- ↑ Darij Grinberg (2003), On the Kosnita Point and the Reflection Triangle. Forum Geometricorum, volume 3, pages 105–111. ISSN 1534-1178
- ↑ John Rigby (1997), Brief notes on some forgotten geometrical theorems. Mathematics and Informatics Quarterly, volume 7, pages 156-158 (as cited by Kimberling).
- ↑ Clark Kimberling (2014), Encyclopedia of Triangle Centers , section X(54) = Kosnita Point. Accessed on 2014-10-08
- ↑ Nikolaos Dergiades (2014), Dao's Theorem on Six Circumcenters associated with a Cyclic Hexagon. Forum Geometricorum, volume 14, pages=243–246. ISSN 1534-1178.
- ↑ Telv Cohl (2014), 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.
- ↑ 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), X(3649) = KS(INTOUCH TRIANGLE)
- ↑ Nguyễn Minh Hà, 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, 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....
- ↑ 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.
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