Question

三角形內角平分線性質定理：三角形的內角平分線分對邊所得的兩條線段和這個角的兩邊對應成比例。 已知：如圖，△ABC中，AD是角平分線． 求證：(1)BD/DC＝AB/AC (2)若AD是三角形ABC外角的平分線,交BC延長線于點D，是否還有以上結論？

Answer 1

(1) 過C作CE∥DA，交BA的延長線于E．∵CE∥DA∴∠1＝∠E，∠2＝∠3，∠1＝∠2∴∠E＝∠3∴AE＝AC∵CE∥DA∴BD/DC＝BA/AE又∵AE＝AC∴BD/DC＝AB/AC(2)在BA延長線上取點C'，使AC'=AC，過C'作C'D'//CD交DA延長線于點D',連接C'D.∵C'D'//CD，A是BA與DD'的交點∴△ABD∽△AC'D'∴BD/C'D'=AB/AC'∵C'D'//CD∴∠C'D'A=∠ADB∵AD是三角形ABC外角的平分線∴∠C'AD=∠CAD∵AC'=AC,AD是公共邊∴△C'AD≌△CAD∴∠C'DA=∠ADB,C'D=CD∴∠C'DA=∠C'D'A∴C'D'=C'D=CD∴BD/DC=BD/C'D'=AB/AC'=AB/AC