Question

三角形內(nèi)角平分線性質(zhì)定理：三角形的內(nèi)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(jié)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