若AD是三角形ABC外角的平分線,交BC延長(zhǎng)線于點(diǎn)D,試根據(jù)相似三角形對(duì)應(yīng)邊成比例說(shuō)明BD/DC=AB/AC

數(shù)學(xué)人氣：171 ℃時(shí)間：2019-10-31 01:18:00

優(yōu)質(zhì)解答

在BA延長(zhǎng)線上取點(diǎn)C',使AC'=AC,過(guò)C'作C'D'//CD交DA延長(zhǎng)線于點(diǎn)D',連接C'D.
∵C'D'//CD,A是BA與DD'的交點(diǎn)
∴△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