如图,已知在△ABC中,点D、E分别在AB、AC上,且AD•AB=AE•AC,CD与BE相交于点O. (1)求证:△AEB∽△ADC; (2)求证:BO/CO=DO/EO.
问题描述:
如图,已知在△ABC中,点D、E分别在AB、AC上,且AD•AB=AE•AC,CD与BE相交于点O.
(1)求证:△AEB∽△ADC;
(2)求证:
=BO CO
. DO EO
答
证明:(1)∵AD•AB=AE•AC,
∴
=AB AC
,AE AD
又∵∠EAB=∠DAC,
∴△AEB∽△ADC;
(2)∵△AEB∽△ADC;
∴∠DBO=∠ECO,
又∵∠DOB=∠EOC,
∴△BOD∽△COE,
∴
=BO CO
.DO EO