コスニタの定理

三角形におけるコスニタの定理（コスニタのていり）は、ある3本の線が共点であるという定理である。

三角形 $ABC$ の外心を $O$ とし、三角形 $OBC$ , 三角形 $OCA$ , 三角形 $OAB$ の外心を $O_{a},O_{b},O_{c}$ とする。このとき「3本の直線 $AO_{a}$ , $BO_{b}$ , $CO_{c}$ は1点で交わる」というのがコスニタの定理である^[1]。この定理の名前はルーマニアの数学者 Cezar Coşniţă（ルーマニア語版）に由来する^[2]。

上記の3本の線の交点はジョン・リグビーによってコスニタ点と命名されている。この点は九点円の中心の等角共役点になっている^[3]^[4]。この点は Encyclopedia of Triangle Centers において $X(54)$ として登録されている^[5]^[6]。この定理はダオの六角形の周上の六円定理（オランダ語版）の特殊な場合である^[7]^[8]^[9]^[10]^[11]^[12]^[13]。

コスニタ点の三線座標は以下の様に与えられる^[6]。

$\sec(B-C):\sec(C-A):\sec(A-B)$

参考文献

^ Weisstein, Eric W. "Kosnita Theorem". mathworld.wolfram.com (英語).
^ 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 Archived 2012-04-19 at the Wayback Machine., section X(54) = Kosnita Point. Accessed on 2014-10-08
^ ^a ^b “ENCYCLOPEDIA OF TRIANGLE CENTERS X54”. faculty.evansville.edu. 2024年3月26日閲覧。
^ 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.

[wolframKosnita-1] Weisstein, Eric W. "Kosnita Theorem". mathworld.wolfram.com (英語).

[patrascu-2] Ion Pătraşcu (2010), A generalization of Kosnita's theorem (in Romanian)

[grinberg2003-3] Darij Grinberg (2003), On the Kosnita Point and the Reflection Triangle. Forum Geometricorum, volume 3, pages 105–111. ISSN 1534-1178

[rigby1997-4] John Rigby (1997), Brief notes on some forgotten geometrical theorems. Mathematics and Informatics Quarterly, volume 7, pages 156-158 (as cited by Kimberling).

[kimberlingX54-5] Clark Kimberling (2014), Encyclopedia of Triangle Centers Archived 2012-04-19 at the Wayback Machine., section X(54) = Kosnita Point. Accessed on 2014-10-08

[:0-6] “ENCYCLOPEDIA OF TRIANGLE CENTERS X54”. faculty.evansville.edu. 2024年3月26日閲覧。

[dergiadesDao6CCCH-7] Nikolaos Dergiades (2014), Dao’s Theorem on Six Circumcenters associated with a Cyclic Hexagon. Forum Geometricorum, volume 14, pages=243–246. ISSN 1534-1178.

[cohlDao6CCCH-8] 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.

[NgoQuangDuong-9] 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

[C.Kimberling-10] Clark Kimberling (2014), X(3649) = KS(INTOUCH TRIANGLE)

[11] 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....

[12] 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....

[13] 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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