Question

已知，如图△ABC中，AD是∠BAC的平分线，AD的垂直平分线交AD于E，交BC延长线于F，求证FD²=FB*FC

Answer 1

连接AF，易得△AEF全等于△DEF，所以AF=DF,∠EAF=∠EDF
∠FAC=∠EAF-∠DAC
∠B=∠DAC-∠DAB
又∠DAB=-∠DAC，所以∠FAC=∠B
所以△ACF等于△BAF
AF/CF=BF/AF
又AF=DF
得证

Answer 2

连接AF，易得△AEF全等于△DEF，所以AF=DF,∠EAF=∠EDF
∠FAC=∠EAF-∠DAC
∠B=∠DAC-∠DAB
又∠DAB=-∠DAC，所以△ACF等于△BAF
AF/CF=BF/AF
AF=DF
得证