C. Kimberling 广泛地列表和列举了三角形中心的性质(Kimberling 1994, 1998 及在线),在他的编号方案中,第
个中心用
表示。Kimberling (1994) 中出现了 101 个(加上额外的 13 个)中心,Kimberling (1998) 中出现了 360 个,其余的中心出现在 Kimberling 在 http://faculty.evansville.edu/ck6/encyclopedia/ETC.html 维护的在线列表中。为了纪念他,这些中心在这项工作中被称为 Kimberling 中心。截至 2004 年 12 月,Kimberling 的汇编包含 3053 个中心。上面展示了其中一部分。
下表总结了前几个 Kimberling 中心,包括它们的编号、名称和三线坐标。
![X_n](/images/equations/KimberlingCenter/Inline3.svg) | 中心 | 三角形中心函数 ![alpha](/images/equations/KimberlingCenter/Inline4.svg) |
![X_1](/images/equations/KimberlingCenter/Inline5.svg) | 内心 ![I](/images/equations/KimberlingCenter/Inline6.svg) | 1 |
![X_2](/images/equations/KimberlingCenter/Inline7.svg) | 三角形重心 ![G](/images/equations/KimberlingCenter/Inline8.svg) | , , ![cscA](/images/equations/KimberlingCenter/Inline11.svg) |
![X_3](/images/equations/KimberlingCenter/Inline12.svg) | 外心 ![O](/images/equations/KimberlingCenter/Inline13.svg) | , ![a(b^2+c^2-a^2)](/images/equations/KimberlingCenter/Inline15.svg) |
![X_4](/images/equations/KimberlingCenter/Inline16.svg) | 垂心 ![H](/images/equations/KimberlingCenter/Inline17.svg) | ![secA](/images/equations/KimberlingCenter/Inline18.svg) |
![X_5](/images/equations/KimberlingCenter/Inline19.svg) | 九点中心 ![N](/images/equations/KimberlingCenter/Inline20.svg) | ,
,
![bc[a^2b^2+a^2c^2-(b^2-c^2)^2]](/images/equations/KimberlingCenter/Inline23.svg) |
![X_6](/images/equations/KimberlingCenter/Inline24.svg) | 外心对称点 ![K](/images/equations/KimberlingCenter/Inline25.svg) | ,
![sinA](/images/equations/KimberlingCenter/Inline27.svg) |
![X_7](/images/equations/KimberlingCenter/Inline28.svg) | 热尔岗点 ![Ge](/images/equations/KimberlingCenter/Inline29.svg) | ,
![sec^2(1/2A)](/images/equations/KimberlingCenter/Inline31.svg) |
![X_8](/images/equations/KimberlingCenter/Inline32.svg) | 内格尔点 ![Na](/images/equations/KimberlingCenter/Inline33.svg) | , ![csc^2(1/2A)](/images/equations/KimberlingCenter/Inline35.svg) |
![X_9](/images/equations/KimberlingCenter/Inline36.svg) | 中点内心 ![M](/images/equations/KimberlingCenter/Inline37.svg) | , ![cot(1/2A)](/images/equations/KimberlingCenter/Inline39.svg) |
![X_(10)](/images/equations/KimberlingCenter/Inline40.svg) | 施皮克中心 ![Sp](/images/equations/KimberlingCenter/Inline41.svg) | ![bc(b+c)](/images/equations/KimberlingCenter/Inline42.svg) |
![X_(11)](/images/equations/KimberlingCenter/Inline43.svg) | 费尔巴哈点 ![F](/images/equations/KimberlingCenter/Inline44.svg) | ,
![sin^2((B-C)/2)](/images/equations/KimberlingCenter/Inline46.svg) |
![X_(12)](/images/equations/KimberlingCenter/Inline47.svg) | 关于 和 的调和共轭点 | , , ![bc(b+c)^2/(b+c-a)](/images/equations/KimberlingCenter/Inline53.svg) |
![X_(13)](/images/equations/KimberlingCenter/Inline54.svg) | 第一费马点 ![X](/images/equations/KimberlingCenter/Inline55.svg) | ,
![sec(A-pi/6)](/images/equations/KimberlingCenter/Inline57.svg) |
![X_(14)](/images/equations/KimberlingCenter/Inline58.svg) | 第二费马点 ![X^'](/images/equations/KimberlingCenter/Inline59.svg) | ,
![sec(A+pi/6)](/images/equations/KimberlingCenter/Inline61.svg) |
![X_(15)](/images/equations/KimberlingCenter/Inline62.svg) | 第一等力点 ![S](/images/equations/KimberlingCenter/Inline63.svg) | ,
![cos(A-pi/6)](/images/equations/KimberlingCenter/Inline65.svg) |
![X_(16)](/images/equations/KimberlingCenter/Inline66.svg) | 第二等力点 ![S^'](/images/equations/KimberlingCenter/Inline67.svg) | ,
![cos(A+pi/6)](/images/equations/KimberlingCenter/Inline69.svg) |
![X_(17)](/images/equations/KimberlingCenter/Inline70.svg) | 第一拿破仑点 ![N](/images/equations/KimberlingCenter/Inline71.svg) | ,
![sec(A-pi/3)](/images/equations/KimberlingCenter/Inline73.svg) |
![X_(18)](/images/equations/KimberlingCenter/Inline74.svg) | 第二拿破仑点 ![N^'](/images/equations/KimberlingCenter/Inline75.svg) | ,
![sec(A+pi/3)](/images/equations/KimberlingCenter/Inline77.svg) |
![X_(19)](/images/equations/KimberlingCenter/Inline78.svg) | 克劳森点 | , , , ![sin(2B)+sin(2C)-sin(2A)](/images/equations/KimberlingCenter/Inline82.svg) |
![X_(20)](/images/equations/KimberlingCenter/Inline83.svg) | 德朗尚点 ![L](/images/equations/KimberlingCenter/Inline84.svg) | ![cosA-cosBcosC](/images/equations/KimberlingCenter/Inline85.svg) |
另请参阅
主要三角形中心,
三角形中心,
三角形中心函数
使用 Wolfram|Alpha 探索
参考文献
Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-167, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Clark Kimberling's Encyclopedia of Triangle Centers--ETC." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html.Kimberling, C. "Encyclopedia of Triangle Centers." http://faculty.evansville.edu/ck6/encyclopedia/.Kimberling, C. "Triangle Centers." http://faculty.evansville.edu/ck6/tcenters/.Pegg, E. Jr. and Weisstein, E. W. "Seven Mathematical Tidbits." MathWorld Headline News. Nov. 8, 2004. https://mathworld.net.cn/news/2004-11-08/seventidbits/#3.在 Wolfram|Alpha 中被引用
Kimberling 中心
请引用为
Weisstein, Eric W. "Kimberling 中心。" 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/KimberlingCenter.html
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