如图,已知AD是△ABC的外角∠EAC的平分线,交BC的延长线于点D,延长DA交△ABC的外接圆于点F,连接FB、FC. (1)求证:FB=FC; (2)求证:FB2=FA•FD;
问题描述:
如图,已知AD是△ABC的外角∠EAC的平分线,交BC的延长线于点D,延长DA交△ABC的外接圆于点F,连接FB、FC.
(1)求证:FB=FC;
(2)求证:FB2=FA•FD;
答
(Ⅰ)∵AD平分∠EAC,
∴∠EAD=∠DAC.
∵四边形AFBC内接于圆,
∴∠DAC=∠FBC.
∵∠EAD=∠FAB=∠FCB,
∴∠FBC=∠FCB,
∴FB=FC.
(Ⅱ)∵∠FAB=∠FCB=∠FBC,∠AFB=∠BFD,
∴△FBA∽△FDB.
∴
=FB FD
,FA FB
∴FB2=FA•FD.