在Rt△ABC中,∠C=90°,BC=2AC,AD是∠BAC的平分线,求证:AB+2BD=5 AC

问题描述:

在Rt△ABC中,∠C=90°,BC=2AC,AD是∠BAC的平分线,求证:AB+2BD=5 AC

不妨设 AC = 1 ,可得:BC = 2 ,AB = √5 .
已知,AD是∠BAC的平分线,
可得:BD/CD = AB/AC = √5 ,
可求得:BD = (1/2)(5-√5) ,CD = (1/2)(√5-1) ,
所以,AB+2BD = 5 = 5AC .