Question

如图,在△ABD和△ACE,AE=AB,AD=AC,∠EAB=∠DAC,BD和CE交于点O,求证：AO平分∠EOD

Answer 1

证明：过点A作AM⊥EC于M,AN⊥BD于N

∵∠EAC＝∠EAB+∠BAC,∠BAD＝∠DAC+∠BAC,∠EAC＝∠DAC

∴∠EAC＝∠BAD

∵AE＝AB,AD＝AC

∴△AEC≌△ABD （SAS）

∴EC＝BD,S△AEC＝S△ABD

∵AM⊥EC,AN⊥BD

∴S△AEC＝EC×AM/2,S△ABD＝BD×AN/2

∴EC×AM/2＝BD×AN/2

∴AM＝AN

∴AO平分∠EOD

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