BEC = 90oР在△BEF和△BEC中Р∠1 = ∠2 Р BE = BEР∠BEF =∠BECР∴△BEF≌△BECР∴CE = FE =CFР∵∠BAC = 90o , BE⊥CFР∴∠BAC = ∠CAF = 90o Р∠1+∠BDA = 90oР∠1+∠BFC = 90oР∠BDA = ∠BFCР在△ABD和△ACF中Р∠BAC = ∠CAFР∠BDA = ∠BFCРAB = ACР∴△ABD≌△ACFР∴BD = CFР∴BD = 2CEР练习:已知,如图,∠ACB = 3∠B,∠1 =∠2,CD⊥AD于D,Р求证:AB-AC = 2CDР规律31.当证题有困难时,可结合已知条件,把图形中的某两点连接起来构造全等三角形.Р例:已知,如图,AC、BD相交于O,且AB = DC,AC = BD,Р求证:∠A = ∠DР证明:(连结BC,过程略)Р规律32.当证题缺少线段相等的条件时,可取某条线段中点,为证题提供条件.Р例:已知,如图,AB = DC,∠A = ∠DР 求证:∠ABC = ∠DCBР 证明:分别取AD、BC中点N、M,Р连结NB、NM、NC(过程略)Р规律33.有角平分线时,常过角平分线上的点向角两边做垂线,利用角平分线上的点到角两边距离相等证题.Р例:已知,如图,∠1 = ∠2 ,P为BN上一点,且PD⊥BC于D,AB+BC = 2BD,Р求证:∠BAP+∠BCP = 180oР证明:过P作PE⊥BA于EР∵PD⊥BC,∠1 = ∠2 Р∴PE = PDР在Rt△BPE和Rt△BPD中РBP = BPРPE = PDР∴Rt△BPE≌Rt△BPDР∴BE = BDР∵AB+BC = 2BD,BC = CD+BD,AB = BE-AEР∴AE = CDР∵PE⊥BE,PD⊥BCР∠PEB =∠PDC = 90oР在△PEA和△PDC中РPE = PD
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