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内切圆

三角形的角平分線會相交於內切圓的圓心

數學中,若一個二維平面上的多邊形的每條邊都能與其內部的一個圓形相切,該圓就是所謂的多邊形的內切圓,這時稱這個多邊形為圓外切多邊形。它亦是多邊形內部最大的圓形。内切圓的圓心被稱為該多邊形的内心

一個多邊形至多有一個内切圓,也就是說對於一個多邊形,它的内切圓,如果存在的話,是唯一的。並非所有的多邊形都有内切圓。三角形正多邊形一定有内切圓。擁有内切圓的四邊形被稱為圆外切四边形

三角形的內切圓

任何三角形都有內切圓。這個內切圓的圓心稱為內心,一般标记为I,是三角形內角平分線的交點[1]。在三線坐標,內心是1:1:1。

性质

內切圓的半徑,當中表示三角形的面積,a、b、c為三角形的三個邊長。

以內切圓和三角形的三個切點為頂點的三角形内接三角形之一。的內切圓就是外接圓。而三线交于一点,它们的交點就是熱爾崗點(Gergonne point)。内切圆与九点圆相切,切点称作费尔巴哈点(见九点圆)。

若以三角形的内切圆为反演圆进行反演,则三角形的三条边和外接圆会分别变为半径相等的四个圆(半径都等于内切圆半径的一半)。[2]

三角形的外接圆半径R、内切圆半径r 以及内外心间距OI 之间有如下关系:

[3]

直角三角形兩股和等於斜邊長加上該三角形內切圓直徑

由此性質再加上勾股定理,可推得:

直角座標系中,若頂點座標分別為,則内心的座標為:

[4]

四边形的内切圆

不是所有的四边形都有内切圆,拥有内切圆的四边形称为圆外切四边形。凸四边形ABCD有内切圆当且仅当两对对边之和相等:。圆外切四边形的面积和内切圆半径的关系为: ,其中s 为半周长。

同时拥有内切圆和外接圆的四边形称为双心四边形。这样的四边形有无限多个。若一个四边形为双心四边形,那么其内切圆在两对对边的切点的连线相互垂直。而只要在一个圆上选取两条相互垂直的,并过相应的顶点做切线,就能得到一个双心四边形。

正多边形的内切圆

正多边形必然有内切圆,而且其内切圆的圆心和外接圆的圆心重合,都在正多边形的中心。边长为a 的正多边形的内切圆半径为:

其内切圆的面积为:

内切圓面積與正多邊形的面積之比為:

故此,當正多邊形的邊數趨向無窮時,

参考文献

  1. ^ R.A.约翰逊,《近代欧氏几何学》,单墫 译,第158页,上海教育出版社,ISBN 7-5320-6392-5
  2. ^ 《近代欧氏几何学》,第163页
  3. ^ 《近代欧氏几何学》,第162页
  4. ^ 平面向量教学与三角形内心. [2013-12-05]. (原始内容存档于2020-08-07). 

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