基本概念
給出△ABC,以頂點A,B,C為圓心,作三個圓兩兩相切,並作與⊙A,⊙B,⊙C都外切的圓⊙S,以及與⊙A,⊙B,⊙C都相內切的圓⊙S'。
⊙S稱為 內索迪(Soddy)圓,⊙S'稱為 外索迪圓,如圖1 。
![圖1](/img/f/df9/wZwpmLygjNxQzM0cDOwcTN1UTM1QDN5MjM5ADMwAjMwUzL3gzL4UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![圖2](/img/d/b83/wZwpmL1ATOzMjNwATMxcTN1UTM1QDN5MjM5ADMwAjMwUzLwEzL3UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![索蒂圓](/img/6/3f6/wZwpmLzEzNxETO1QDMxcTN1UTM1QDN5MjM5ADMwAjMwUzL0AzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
英國的化學家索迪(Frederick Soddy,1877-1958)求出了這些圓的半徑 ,它們是
![索蒂圓](/img/b/865/wZwpmLwUTN5MzN5QDMxcTN1UTM1QDN5MjM5ADMwAjMwUzL0AzLwEzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
事實上,這些結果在17世紀已經被笛卡爾發現 。
索迪線
內索迪圓圓心S和外索迪圓圓心S'的連線稱為 索迪線,由於內心和熱爾崗點也在此直線上,故也把聯結I和熱爾崗點Ge的直線稱為索迪線。它的三線方程是
![索蒂圓](/img/b/88e/wZwpmL4IzN5EDM0gDMxcTN1UTM1QDN5MjM5ADMwAjMwUzL4AzL2EzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
索迪線上S',I,S和Ge成調和點列,索迪線垂直於熱爾崗線。
相關性質介紹
一個三角形ABC的歐拉線,上面有6個著名的點:垂心H,九點圓圓心N,重心G,外心O,德朗謝姆點Z,以及三角形的外接圓在A,B,C的切線所成三角形的外心。德朗謝姆點與H關於O對稱,是外接圓與斯坦納圓(以N為心,3R/2為半徑)的外相似中心(在斯坦納圓內,一個半徑為R/2的圓沿圓周滾動時,畫出有3個尖點的圓內旋輪線,它是西摩松線的包絡)。
採用通常的記號,以任一三角形ABC的頂點為圓心,s-a,s-b,s-c,或者s,s-c,s-b,或者s-c,s,s-a,或者s-b,s-a,s為半徑所作的3個圓兩兩相切。第一種情況如圖1所示,在其餘的每一種情況,一個圓包住另外兩個,對這3個圓的每一種情況,都可以找到另外兩個圓與它們都相切,稱這兩個圓為三角形的 索迪圓,並將它們的圓心與半徑記為
![索蒂圓](/img/7/385/wZwpmL3UzM0QTM5QDOwcTN1UTM1QDN5MjM5ADMwAjMwUzL0gzL2gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![索蒂圓](/img/d/c03/wZwpmL2EzN4ADM4ITOwcTN1UTM1QDN5MjM5ADMwAjMwUzLykzLyUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![索蒂圓](/img/2/47e/wZwpmLyIjM4UDO3cDMxcTN1UTM1QDN5MjM5ADMwAjMwUzL3AzLwczLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
其中σ< σ',等等,我們將發現這四對圓心落在4條直線上,其中是內心與旁心,而Z是德朗謝姆點 。
![圖3](/img/7/a45/wZwpmLyIzMxYDOxgjMxcTN1UTM1QDN5MjM5ADMwAjMwUzL4IzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![圖4 ( 圖3的放大圖)](/img/8/e02/wZwpmL1IjN0MTN1IDMxcTN1UTM1QDN5MjM5ADMwAjMwUzLyAzL0AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
在4R+r<2s時,第一對索迪圓的位置如圖1,令L為BC中點,X,X' ,M,K為S,S',I,Z在BC上的射影,容易得到關係
![索蒂圓](/img/6/66c/wZwpmL3AzM5kzM2YDMxcTN1UTM1QDN5MjM5ADMwAjMwUzL2AzLwczLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![索蒂圓](/img/c/731/wZwpmLwQDNxkzNxADMxcTN1UTM1QDN5MjM5ADMwAjMwUzLwAzLwEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![索蒂圓](/img/2/2d5/wZwpmL3IDO3IzM0MDMxcTN1UTM1QDN5MjM5ADMwAjMwUzLzAzL4AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
因此
![索蒂圓](/img/e/e99/wZwpmL4EjNyADO5kDMxcTN1UTM1QDN5MjM5ADMwAjMwUzL5AzL4czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![索蒂圓](/img/c/e67/wZwpmL0IzM1cTO4ETOwcTN1UTM1QDN5MjM5ADMwAjMwUzLxkzL2EzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
因此I是索迪圓的內相似中心,S與S'在直線IZ上,Z分SS'為比(σ+s):(σ'-s),並且
![索蒂圓](/img/f/cb5/wZwpmL2AzNxEDO2MTMxcTN1UTM1QDN5MjM5ADMwAjMwUzLzEzLxQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![索蒂圓](/img/1/e4d/wZwpmL1gTM5QTN3ETOwcTN1UTM1QDN5MjM5ADMwAjMwUzLxkzL2MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
容易算出,由此得出SS'的表達式,同時這也順便證明了
![索蒂圓](/img/d/6d1/wZwpmLzUDO3cjMwgDMxcTN1UTM1QDN5MjM5ADMwAjMwUzL4AzL2AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
(根據O.Bottema給H.S.M.Coxeter 的信,G.R.Veldkamp已經注意到外相似中心是約爾剛點,AM在這裡與自B,C引出的類似的塞瓦線相交)
在4R +r>2s時,類似的計算表明J是內相似中心,Z外分SS',比為(σ+s):(σ'+s),並且
![索蒂圓](/img/1/623/wZwpmL0QzM4EzNygDMxcTN1UTM1QDN5MjM5ADMwAjMwUzL4AzL0gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![索蒂圓](/img/5/5e9/wZwpmL1QjM4IzM5UDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL1gzL1IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![索蒂圓](/img/3/ea4/wZwpmLyITMzgDNyYTMxcTN1UTM1QDN5MjM5ADMwAjMwUzL2EzLwEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![索蒂圓](/img/3/ea4/wZwpmLyITMzgDNyYTMxcTN1UTM1QDN5MjM5ADMwAjMwUzL2EzLwEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![索蒂圓](/img/7/c37/wZwpmL3QDM5QTM4YDMxcTN1UTM1QDN5MjM5ADMwAjMwUzL2AzL4EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![索蒂圓](/img/6/31b/wZwpmL2cjN3MDMzMDMxcTN1UTM1QDN5MjM5ADMwAjMwUzLzAzLyIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
對餘下的三對索迪圓,我們必須用旁心代替內心。例如,我們發現是圓心為的索迪圓的外相似中心。在直線上,Z內分比為(s-a-σ):(σ'-s+a)。
![索蒂圓](/img/6/31b/wZwpmL2cjN3MDMzMDMxcTN1UTM1QDN5MjM5ADMwAjMwUzLzAzLyIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![索蒂圓](/img/d/b69/wZwpmL0MTMxQjN5IDOwcTN1UTM1QDN5MjM5ADMwAjMwUzLygzL0EzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![索蒂圓](/img/8/5e0/wZwpmLzATN2YTMxADMxcTN1UTM1QDN5MjM5ADMwAjMwUzLwAzLwAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![索蒂圓](/img/5/6c0/wZwpmL0QTO1QDOzUDOwcTN1UTM1QDN5MjM5ADMwAjMwUzL1gzL2YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
圖5表明歐拉線HO與四條“索迪線"SS',,,都過Z點,為了避免圖形的複雜性,我們省掉了直線,雖然它們也是有些趣味的,因為它們是由內切圓與3個旁切圓在BC,CA,AB上的切點所成的4個三角形的歐拉線 。
![圖5](/img/e/08d/wZwpmLwAzN3EDOycTMxcTN1UTM1QDN5MjM5ADMwAjMwUzL3EzLxAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)