向量形式
公式
![圖一](/img/7/1e2/wZwpmL4czM3YDN3kjNxMzM1UTM1QDN5MjM5ADMwAjMwUzL5YzL4AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![定比分點公式](/img/9/cb5/wZwpmL4QjNyMjM2MzM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzMzLxIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
設D為 一邊BC上一點,則
![定比分點公式](/img/6/117/wZwpmLwMjMyEzNzYTMxMzM1UTM1QDN5MjM5ADMwAjMwUzL2EzLwYzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![定比分點公式](/img/c/8da/wZwpmL1MjNwcTNygjNwMzM1UTM1QDN5MjM5ADMwAjMwUzL4YzLwIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
設 ,則
![定比分點公式](/img/5/8e2/wZwpmL0cTMwITOzEzNxMzM1UTM1QDN5MjM5ADMwAjMwUzLxczL4MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
證明過程
對於前者,
![定比分點公式](/img/6/f99/wZwpmL1IDO2kDM4kjNwMzM1UTM1QDN5MjM5ADMwAjMwUzL5YzL3AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![定比分點公式](/img/9/cc0/wZwpmL0MDO5gDN0gjNwMzM1UTM1QDN5MjM5ADMwAjMwUzL4YzL3EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
對於後者,將 和
![定比分點公式](/img/1/d74/wZwpmLzcDMxQDNwczNxMzM1UTM1QDN5MjM5ADMwAjMwUzL3czLxczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![定比分點公式](/img/c/8da/wZwpmL1MjNwcTNygjNwMzM1UTM1QDN5MjM5ADMwAjMwUzL4YzLwIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
相加,套用 即得結論。
坐標形式
公式
![定比分點公式](/img/0/82a/wZwpmL2cTM1AjNycTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzLxYzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![定比分點公式](/img/8/a7a/wZwpmLzMTN3ATO2czM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3MzLxUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![定比分點公式](/img/d/423/wZwpmLyEDN1UjMyUjNxIDN0UTMyITNykTO0EDMwAjMwUzL1YzLygzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![定比分點公式](/img/5/b10/wZwpmL4UjNyMDN4AzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczLzQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![定比分點公式](/img/0/12c/wZwpmL3EDM3kTOyUTMxMzM1UTM1QDN5MjM5ADMwAjMwUzL1EzL4UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![定比分點公式](/img/3/2f8/wZwpmLwIDNxAjMxMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLyIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
在平面直角坐標系內,已知兩點 , ;在兩點連線上有一點P,設它的坐標為 ,且 ,那么我們說P分有向線段 的比為 。我們將
![定比分點公式](/img/2/88e/wZwpmL1MDM2QjMzkjNwMzM1UTM1QDN5MjM5ADMwAjMwUzL5YzLzIzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![定比分點公式](/img/f/8a0/wZwpmL4UjMxETNygjMxMzM1UTM1QDN5MjM5ADMwAjMwUzL4IzLxEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![定比分點公式](/img/2/d14/wZwpmLxcTN1ADM5QDOwMzM1UTM1QDN5MjM5ADMwAjMwUzL0gzL3QzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![定比分點公式](/img/2/518/wZwpmL2MzMxkTOyUTMxMzM1UTM1QDN5MjM5ADMwAjMwUzL1EzL0IzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
稱為有向線段 的定比分點P的坐標公式。
![定比分點公式](/img/0/71f/wZwpmL1gTO0ETO0cDMyMzM1UTM1QDN5MjM5ADMwAjMwUzL3AzL2EzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![定比分點公式](/img/3/c43/wZwpmL0QzN4ADNxATOxMzM1UTM1QDN5MjM5ADMwAjMwUzLwkzL2YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![定比分點公式](/img/a/8b2/wZwpmLyUTO2MDN1MzMxMzM1UTM1QDN5MjM5ADMwAjMwUzLzMzL4UzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![定比分點公式](/img/3/2f8/wZwpmLwIDNxAjMxMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLyIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
當P為內分點時, ;當P為外分點時, ;當P與A重合時, ;當P與B重合時, 不存在。
推導過程
![定比分點公式](/img/5/133/wZwpmLwETOwUDN2ETOxMzM1UTM1QDN5MjM5ADMwAjMwUzLxkzLwEzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
對 及其上一點P套用定比分點公式的向量形式,即證。
推論
下面不加證明地給出兩個推論。
中點公式
![定比分點公式](/img/0/82a/wZwpmL2cTM1AjNycTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3EzLxYzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![定比分點公式](/img/8/a7a/wZwpmLzMTN3ATO2czM2EzM1UTM1QDN5MjM5ADMwAjMwUzL3MzLxUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![定比分點公式](/img/0/f99/wZwpmL3YTM4EDO5YTM0EDN0UTMyITNykTO0EDMwAjMwUzL2EzL4czLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
已知兩點 , ,設兩點中點為 ,則推出中點公式
![定比分點公式](/img/c/1fa/wZwpmLxEjN0MDM0ITOxMzM1UTM1QDN5MjM5ADMwAjMwUzLykzL3AzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![定比分點公式](/img/f/4ce/wZwpmL4QDMwcTN4gDOxMzM1UTM1QDN5MjM5ADMwAjMwUzL4gzLwUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
三角形重心公式
![定比分點公式](/img/d/889/wZwpmL0MjN3ATNyAjNxMzM1UTM1QDN5MjM5ADMwAjMwUzLwYzLxczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![定比分點公式](/img/d/1f3/wZwpmLwUDNwgDMxADMyMzM1UTM1QDN5MjM5ADMwAjMwUzLwAzL4MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
已知 ,設三角形重心為 ,則推出三角形重心公式
![定比分點公式](/img/1/d96/wZwpmL0MDMzYDOzczMxMzM1UTM1QDN5MjM5ADMwAjMwUzL3MzLwAzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![定比分點公式](/img/7/af8/wZwpmL3MjMwETO4QDOwMzM1UTM1QDN5MjM5ADMwAjMwUzL0gzL1YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)