= ∠ANCР∵∠B+∠ACB+∠ACN+∠ANC = 180oР∴2∠BCA+2∠ACN = 180oР∴∠BCA+∠ACN = 90oР即∠BCN = 90oР∴NC⊥BCР∵AE = AFР∴∠AEF = ∠AFEР又∵∠BAC = ∠AEF +∠AFEР∠BAC = ∠ACN +∠ANCР∴∠BAC =2∠AEF = 2∠ANCР∴∠AEF = ∠ANCР∴EF∥NCР∴EF⊥BCР⑷常过一腰上的某一已知点做另一腰的平行线Р例:已知,如图,在△ABC中,AB = AC,D在AB上,E在AC延长线上,且BD = CE,连结DE交BC于FР求证:DF = EFР证明:(证法一)过D作DN∥AE,交BC于N,则∠DNB = ∠ACB,Р∠NDE = ∠E,Р∵AB = AC,Р∴∠B = ∠ACBР∴∠B =∠DNBР∴BD = DNР又∵BD = CE Р∴DN = ECР在△DNF和△ECF中Р∠1 = ∠2Р∠NDF =∠EРDN = EC Р∴△DNF≌△ECFР∴DF = EFР(证法二)过E作EM∥AB交BC延长线于M,则∠EMB =∠B(过程略)Р⑸常过一腰上的某一已知点做底的平行线Р例:已知,如图,△ABC中,AB =AC,E在AC上,D在BA延长线上,且AD = AE,连结DEР求证:DE⊥BCР证明:(证法一)过点E作EF∥BC交AB于F,则Р∠AFE =∠BР∠AEF =∠CР∵AB = ACР∴∠B =∠CР∴∠AFE =∠AEFР∵AD = AEР∴∠AED =∠ADEР又∵∠AFE+∠AEF+∠AED+∠ADE = 180oР∴2∠AEF+2∠AED = 90o Р即∠FED = 90o Р∴DE⊥FEР又∵EF∥BCР∴DE⊥BCР(证法二)过点D作DN∥BC交CA的延长线于N,(过程略)Р(证法三)过点A作AM∥BC交DE于M,(过程略)
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