查看“︁雞爪定理”︁的源代码

[[欧几里得几何|歐氏幾何]]中，'''雞爪定理'''<ref>{{cite book|title = 鸡爪定理|author = 金磊 |publisher = [[哈尔滨工业大学出版社]]|year = 2020 |isbn = 9787560384245 |series = 《數學中的小問題大定理》叢書（第六輯）}}</ref>（或'''內心/旁心引理'''，{{lang-en|incenter/excenter lemma}}）描述[[三角形]]的[[頂點 (幾何)|頂點]]、[[內心]]、[[旁心]]、[[外接圓]]的位置關係。其斷言，三角形某頂點<math>B</math>所對的旁心<math>I_B</math>、另兩個頂點<math>A</math>、<math>C</math>、內心<math>I</math>[[四點共圓]]，且其[[圓心]]（<math>II_B</math>中點）位於三角形的外接圓<math>\odot ABC</math>上。此定理的構形常於[[數學競賽|奧數]]幾何題出現。<ref name ="egmo">{{cite book|url = https://www.google.co.uk/books/edition/Euclidean_Geometry_in_Mathematical_Olymp/47UaDAAAQBAJ?hl=en&gbpv=1&pg=PA9&printsec=frontcover|title = Euclidean Geometry in Mathematical Olympiads|trans-title = 奧數歐氏幾何|author = Evan Chen|isbn = 9780883858394|publisher = [[Mathematical Association of America]]|year = 2016|page = 9–10|quote = This configuration shows up very often in olympiad geometry, so recognize it when it appears!|language = en|access-date = 2021-12-12|archive-date = 2021-12-15|archive-url = https://web.archive.org/web/20211215193544/https://www.google.co.uk/books/edition/Euclidean_Geometry_in_Mathematical_Olymp/47UaDAAAQBAJ?hl=en&gbpv=1&pg=PA9&printsec=frontcover|dead-url = no}}</ref>

== 敍述 ==
[[File:Trillium theorem.svg|thumb|upright=1.2|雞爪定理：三條紅色線段等長]]
設<math>\triangle ABC</math>為[[任意三角形]]，<math>I</math>為其[[內心]]，<math>\angle ABC</math>的[[角平分線]]<math>BI</math>交[[外接圓]]<math>\odot ABC</math>於<math>D</math>。定理斷言，<math>D</math>到<math>A, C, I</math>三點等远，即<math>DA = DC = DI</math>。

等價的說法有：
*過<math>A, C, I</math>三點的圓，圓心位於<math>D</math>。這尤其說明該圓的圓心在於原三角形的外接圓上。<ref>{{citation
 | last = Morris | first = Richard
 | issue = 2
 | journal = {{link-en|數學教師|The Mathematics Teacher|The Mathematics Teacher}}
 | jstor = 27951001
 | pages = 63–71
 | title = Circles through notable points of the triangle
 | trans-title = 過三角形特殊點的圓
 | volume = 21
 | year = 1928| doi = 10.5951/MT.21.2.0069
 | language = en
 }}. 尤其見p.&nbsp;65處關於諸圓<math>BIC, CIA, AIB</math>及圓心的討論。</ref><ref>{{citation|url=http://www.cut-the-knot.org/m/Geometry/OnePropertyOfCircleThroughIncenter.shtml|title=A Property of Circle Through the Incenter|trans-title=過內心的圓之某性質|work=[[Cut-the-Knot]]|author={{le|Alexander Bogomolny|Alexander Bogomolny|Bogomolny, Alexander}}|access-date=2016-01-26|language=en|archive-date=2021-12-12|archive-url=https://web.archive.org/web/20211212155653/http://www.cut-the-knot.org/m/Geometry/OnePropertyOfCircleThroughIncenter.shtml|dead-url=no}}.</ref>
*諸三角形<math>AID, CID, ACD</math>皆為[[等腰三角形|等腰]]，<math>D</math>為其頂角。

還有第四點<math>I_B</math>也到<math>D</math>等遠，就是<math>B</math>所對的[[旁心]]。在以<math>D</math>為圓心的圓上，<math>I_B</math>與<math>I</math>互為[[對徑點]]，即<math>D</math>為<math>II_B</math>的[[中點]]。<ref name="approaching-2014-10-29-inscribed-angles" /><ref>{{citation|url=http://www.cut-the-knot.org/Curriculum/Geometry/InExCircum.shtml|title=Midpoints of the Lines Joining In- and Excenters|trans-title=內心與諸旁心所連線段之中點|work=[[Cut-the-Knot]]|first=Alexander|last=Bogomolny|access-date=2016-01-26|language=en|archive-date=2021-12-12|archive-url=https://web.archive.org/web/20211212155541/http://www.cut-the-knot.org/Curriculum/Geometry/InExCircum.shtml|dead-url=no}}.</ref>

== 證明 ==
由於同弧所對的[[圆周角]]相等，有
: <math>\angle IBA = \angle DCA, \quad \angle IBC = \angle DAC. </math>

又<math>BI</math>為角<math>B</math>的平分線，有
: <math> \angle DCA = \angle DAC,</math>
故<math>DA = DC </math>得證（等圓周角對等[[弦 (幾何)|弦]]）。

最後計角有：
: <math>
\begin{align}
 \angle DIA = {} & 180^\circ - \angle AIB \\
= {} & \angle IAB + \angle IBA \\ 
= {} & \angle IAC + \angle DAC \\
= {} & \angle IAD. \\
\end{align}
</math>
所以三角形<math>DIA</math>有兩底角相等，證畢<math>DA = DI</math>。

==應用於求作三角形==
定理適用於解決以下問題：已知某三角形的一個頂點<math>B</math>、[[內心]]<math>I</math>和[[外心]]<math>O</math>，求作該三角形。作法如下：

#以<math>O</math>為圓心，<math>OB</math>為半徑，作圓。此為三角形的外接圓。
#作直線<math>BI</math>，與外接圓交於（<math>B</math>以外的另一點）<math>D</math>。
#以<math>D</math>為圓心，<math>DI</math>為半徑作圓，定理保證所得的圓過另兩個頂點<math>A, C</math>。
#所以，該圓與外接圓的交點<math>A, C</math>即為所求。<ref>{{citation|title=Problems and Solutions in Euclidean Geometry|trans-title=歐氏幾何問題及解答|series=Dover Books on Mathematics|first1=M. N.|last1=Aref|first2=William|last2=Wernick|publisher=Dover Publications, Inc.|year=1968|isbn=9780486477206|at=3.3(i), p.&nbsp;68|url=https://books.google.com/books?id=vAcU7jOFhG4C&pg=PA68|language=en|accessdate=2021-12-12|archive-date=2021-12-12|archive-url=https://web.archive.org/web/20211212165218/https://books.google.com/books?id=vAcU7jOFhG4C&pg=PA68|dead-url=no}}.</ref>
然而，並非在平面上任意取三點作為<math>B, I, O</math>皆有對應的三角形。若以上作法不能給出三角形，則問題可能出在<math>IB</math>與<math>\odot O</math>相切，也可能在於最後兩圓[[相切]]或[[外離]]。而且，若<math>B, I, O</math>三點無任何限制，則即使作法確實給出三角形，<math>I</math>亦不必為其內心，可能是旁心。該些情況下，不存在三角形以<math>B</math>為頂點，<math>I</math>為內心、<math>O</math>為外心。（對於固定的<math>B, O</math>兩點，若要存在此種三角形，則<math>I</math>必須位於以<math>B</math>為尖點關於<math>\odot O</math>作成的[[心臟線]]圍成的區域中。）<ref name="yiu">{{citation
 | last = Yiu
 | first = Paul
 | issue = 2
 | journal = Journal for Geometry and Graphics
 | mr = 3088369
 | pages = 171–183
 | title = Conic construction of a triangle from its incenter, nine-point center, and a vertex
 | trans-title = 給定內心、九點圓心、一頂點，以圓錐曲線構作原三角形
 | url = http://math.fau.edu/Yiu/j16h2yiu.pdf
 | volume = 16
 | year = 2012
 | language = en
 | accessdate = 2021-12-12
 | archive-date = 2020-11-28
 | archive-url = https://web.archive.org/web/20201128081834/http://math.fau.edu/Yiu/j16h2yiu.pdf
 | dead-url = no
 }}</ref>

其他構作三角形的問題，如給定頂點、內心、[[九點圓|九點圓心]]，求作三角形，有部分情況可化歸為前述問題解決，但一般而言無法[[尺規作圖|尺規作出]]。<ref name="yiu"/>

== 命名 ==
本定理有許多不同的名稱。「雞爪定理」得名自<math>DA, DI, DC</math>諸線段組成的幾何圖形。同樣，俄文稱為{{lang|rus|лемма о трезубце}}<ref name="rkarasev">{{cite book
 |title = Задачи для школьного математического кружка
 |trans-title = 數學興趣小組題目
 |author1 = Р. Н. Карасёв
 |author2 = В. Л. Дольников
 |author3 = И. И. Богданов
 |author4 = А. В. Акопян
 |pages = 4
 |location = Problem 1.2
 |url = http://www.rkarasev.ru/common/upload/taskprob.pdf
 |language = ru
 |access-date = 2021-12-12
 |archive-date = 2021-12-12
 |archive-url = https://web.archive.org/web/20211212155509/http://www.rkarasev.ru/common/upload/taskprob.pdf
 |dead-url = no
 }}</ref><!-- Triangle ABC is inscribed in circle S. I is the center of inscribed circle of the triangle ABC. The line AI intersects S second time in the point D. Prove that DB = DC = DI. --><ref name="approaching-2014-10-29-inscribed-angles">{{cite web
 |url = http://math.mosolymp.ru/upload/files/2015/aesc/approaching-2014-10-29-inscribed-angles.pdf
 |publisher = СУНЦ МГУ им. М. В. Ломоносова - школа им. А.Н. Колмогорова
 |title = 6. Лемма о трезубце
 |trans-title = 6. 三叉引理
 |date = 2014-10-29
 |language = ru
 |access-date = 2021-12-12
 |archive-date = 2021-12-12
 |archive-url = https://web.archive.org/web/20211212155523/https://math.mosolymp.ru/upload/files/2015/aesc/approaching-2014-10-29-inscribed-angles.pdf
 |dead-url = no
 }}</ref>，謂[[三叉]]引理，或{{lang|rus|теорема трилистника}}<ref name="9pointcircle">{{cite journal
 |url = http://www.geometry.ru/persons/kushnir/9pointcircle.pdf
 |title = Это открытие - золотой ключ Леонарда Эйлера
 |trans-title = 這個發現——萊昂哈德·歐拉的金鑰
 |location = Ф7 (Теорема трилистника), p.34；證明見p.36
 |author = И. А. Кушнир
 |access-date = 2021-12-12
 |archive-date = 2021-12-12
 |archive-url = https://web.archive.org/web/20211212155516/https://www.geometry.ru/persons/kushnir/9pointcircle.pdf
 |dead-url = no
 }}</ref>，謂[[三葉草]]定理。英文又稱{{lang|en|theorem of trillium}}「[[延齡草]]定理」，亦是以某種三葉植物命名。

定理亦有其他名稱並非來自該形狀，如「內心/旁心引理」（{{lang|en|the incenter/excenter lemma}}）。<ref name ="egmo"/>

== 參考文獻 ==
{{reflist|30em}}

[[Category:三角形几何]]
[[Category:幾何定理]]