如图,AD为△ABC的角平分线,M为BC的中点,ME‖AD交BA的延长线于E,交AC于F。求证BE=CF=½(AB+AC)
来源:百度知道 编辑:UC知道 时间:2024/06/16 11:20:14
AD为△ABC的角平分线, AC/AB= CD/BD
M为BC的中点, AC/AB= (BM+DM)/(BM-DM)
(AC+AB)/AB = 2BM / (BM-DM)
ME‖AD交BA的延长线于E, △EBM ∽ △ABD
BF/AB = BM/BD = BM / (BM-DM) = ½(AC+AB)/AB
BF =½(AB+AC)
ME‖AD交BA的延长线于E,△CFM ∽ △CAD
CF/AC = CM/CD = BM/(BM+MD)
BF/AC = (BF/AB) * (AB/AC)
=[ BM / (BM-DM)] * [(BM-DM)/(BM+DM)]
= BM/(BM+MD)
= CF/AC
BF = CF
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