如图,点A,B,C,D在⊙O上,AB=AC,AD与BC相交于点E,AE= 1/2 ED,延长DB到点F,使FB= 1/2使FB= 1/2BD 连接AF (1)求证△ABE∽ADB(2)求证直线AF与圆O相切
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![如图,点A,B,C,D在⊙O上,AB=AC,AD与BC相交于点E,AE= 1/2 ED,延长DB到点F,使FB= 1/2使FB= 1/2BD 连接AF (1)求证△ABE∽ADB(2)求证直线AF与圆O相切](/uploads/image/z/7197850-10-0.jpg?t=%E5%A6%82%E5%9B%BE%2C%E7%82%B9A%2CB%2CC%2CD%E5%9C%A8%E2%8A%99O%E4%B8%8A%2CAB%3DAC%2CAD%E4%B8%8EBC%E7%9B%B8%E4%BA%A4%E4%BA%8E%E7%82%B9E%2CAE%3D+1%2F2+ED%2C%E5%BB%B6%E9%95%BFDB%E5%88%B0%E7%82%B9F%2C%E4%BD%BFFB%3D+1%2F2%E4%BD%BFFB%3D+1%2F2BD+%E8%BF%9E%E6%8E%A5AF+%EF%BC%881%EF%BC%89%E6%B1%82%E8%AF%81%E2%96%B3ABE%E2%88%BDADB%EF%BC%882%EF%BC%89%E6%B1%82%E8%AF%81%E7%9B%B4%E7%BA%BFAF%E4%B8%8E%E5%9C%86O%E7%9B%B8%E5%88%87)
如图,点A,B,C,D在⊙O上,AB=AC,AD与BC相交于点E,AE= 1/2 ED,延长DB到点F,使FB= 1/2使FB= 1/2BD 连接AF (1)求证△ABE∽ADB(2)求证直线AF与圆O相切
如图,点A,B,C,D在⊙O上,AB=AC,AD与BC相交于点E,AE= 1/2 ED,延长DB到点F,使FB= 1/2
使FB= 1/2BD 连接AF
(1)求证△ABE∽ADB
(2)求证直线AF与圆O相切
如图,点A,B,C,D在⊙O上,AB=AC,AD与BC相交于点E,AE= 1/2 ED,延长DB到点F,使FB= 1/2使FB= 1/2BD 连接AF (1)求证△ABE∽ADB(2)求证直线AF与圆O相切
证明:(1)∵AB=AC
∴△ABC是等腰三角形,∠ABC=∠ACB
∵点A、B、C、D在⊙O上
∴∠ACB与∠ADB是园周角且同弧AB
∴∠ACB=∠ADB,即∠ABC=∠ADB
∵在△ABE和△ADB中,∠ABC=∠ADB,∠BAD=∠DAB
∴△ABE∽△ADB
(2)连接OA
∵点A、B、C、D在⊙O上,AB=AC
∴OA垂直平分BC
∵AE=1/2ED,FB=1/2BD
∴AD=3/2ED,DF=3/2BD
即AD/ED=DF/BD=3/2
∵在△DAF和△DEB中,
∠ADF=∠EDB,AD/ED=DF/BD
∴△DAF∽△DEB
∴∠F=∠EBD
∴BE∥FA
∴OA⊥AF
∵OA是⊙O的半径
∴AF是⊙O的切线
(2)证明:连接OA,OB,OC,
∵AB=AC,BO=CO,OA=OA,
∴△OAB≌OAC,
∴∠OAB=∠OAC,
∴AO是等腰三角形ABC顶角∠BAC的平分线,
∴AO⊥BC,
∵△BDE∽FDA,
得∠EBD=∠AFD,
∴BE∥FA,
∵AO⊥BE知,AO⊥FA,
∴直线AF与⊙O相切。