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

2个回答

  • 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