1、有等腰三角形时常用的辅助线作顶角的平分线,底边中线,底边高线例:已知,如图,AB = AC,BDAC于D,求证:BAC = 2DBC证明:(方法一)作BAC的平分线AE,交BC于E,则1 = 2 = BAC又AB = ACAEBC2ACB = 90oBDACDBCACB = 90o2 = DBCBAC = 2DBC(方法二)过A作AEBC于E(过程略)(方法三)取BC中点E,连结AE(过程略)有底边中点时,常作底边中线例:已知,如图,ABC中,AB = AC,D为BC中点,DEAB于E,DFAC于F,求证:DE = DF证明:连结AD.D为BC中点,BD = CD又AB =ACAD平分BACD
2、EAB,DFACDE = DF将腰延长一倍,构造直角三角形解题例:已知,如图,ABC中,AB = AC,在BA延长线和AC上各取一点E、F,使AE = AF,求证:EFBC证明:延长BE到N,使AN = AB,连结CN,则AB = AN = ACB = ACB, ACN = ANCBACBACNANC = 180o2BCA2ACN = 180oBCAACN = 90o即BCN = 90oNCBCAE = AFAEF = AFE又BAC = AEF AFEBAC = ACN ANCBAC =2AEF = 2ANCAEF = ANCEFNCEFBC常过一腰上的某一已知点做另一腰的平行线例:已知,如
3、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F求证:DF = EF证明:(证法一)过D作DNAE,交BC于N,则DNB = ACB,NDE = E,AB = AC,B = ACBB =DNBBD = DN又BD = CE DN = EC在DNF和ECF中1 = 2NDF =EDN = EC DNFECFDF = EF(证法二)过E作EMAB交BC延长线于M,则EMB =B(过程略)常过一腰上的某一已知点做底的平行线例:已知,如图,ABC中,AB =AC,E在AC上,D在BA延长线上,且AD = AE,连结DE求证:DEBC证明:(证法一)过点
4、E作EFBC交AB于F,则AFE =BAEF =CAB = ACB =CAFE =AEFAD = AEAED =ADE又AFEAEFAEDADE = 180o2AEF2AED = 90o 即FED = 90o DEFE又EFBCDEBC(证法二)过点D作DNBC交CA的延长线于N,(过程略)(证法三)过点A作AMBC交DE于M,(过程略)常将等腰三角形转化成特殊的等腰三角形-等边三角形例:已知,如图,ABC中,AB = AC,BAC = 80o ,P为形内一点,若PBC = 10o PCB = 30o 求PAB的度数.解法一:以AB为一边作等边三角形,连结CE则BAE =ABE = 60oAE
5、 = AB = BEAB = ACAE = AC ABC =ACBAEC =ACEEAC =BACBAE = 80o 60o = 20oACE = (180oEAC)= 80oACB= (180oBAC)= 50oBCE =ACEACB = 80o50o = 30oPCB = 30oPCB = BCEABC =ACB = 50o, ABE = 60oEBC =ABEABC = 60o50o =10oPBC = 10oPBC = EBC在PBC和EBC中PBC = EBCBC = BCPCB = BCEPBCEBCBP = BEAB = BEAB = BPBAP =BPAABP =ABCPBC
6、= 50o10o = 40oPAB = (180oABP)= 70o解法二:以AC为一边作等边三角形,证法同一。解法三:以BC为一边作等边三角形BCE,连结AE,则EB = EC = BC,BEC =EBC = 60oEB = ECE在BC的中垂线上同理A在BC的中垂线上EA所在的直线是BC的中垂线EABCAEB = BEC = 30o =PCB由解法一知:ABC = 50oABE = EBCABC = 10o =PBCABE =PBC,BE = BC,AEB =PCBABEPBCAB = BP BAP =BPAABP =ABCPBC = 50o10o = 40oPAB = (180oABP)
7、 = (180o40o)= 70o有等腰三角形时常用的辅助线作顶角的平分线,底边中线,底边高线例:已知,如图,AB = AC,BDAC于D,求证:BAC = 2DBC有底边中点时,常作底边中线例:已知,如图,ABC中,AB = AC,D为BC中点,DEAB于E,DFAC于F,求证:DE = DF将腰延长一倍,构造直角三角形解题例:已知,如图,ABC中,AB = AC,在BA延长线和AC上各取一点E、F,使AE = AF,求证:EFBC常过一腰上的某一已知点做另一腰的平行线例: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F求证:DF = EF常过一腰上的某一已知点做底的平行线例:已知,如图,ABC中,AB =AC,F在AC上,E在BA延长线上,且AE = AF,连结DE求证:EFBC常将等腰三角形转化成特殊的等腰三角形-等边三角形例:已知,如图,ABC中,AB = AC,BAC = 80o ,P为形内一点,若PBC = 10o , PCB = 30o 求PAB的度数.5