如图,在Rt△ABC中,AB=AC,∠BAC=90°,BF平分∠ABC,CD⊥BF交BF的延长线于点D.求证：BF=2CD.

Question

如图,在Rt△ABC中,AB=AC,∠BAC=90°,BF平分∠ABC,CD⊥BF交BF的延长线于点D.
求证：BF=2CD.

Answer 1

延长CD并交BA延长线于E
∵BD⊥CE ∠CBD=∠EBD
∴∠BCE=∠E
∴BC=BE
∴CE=2CD
∵∠ABF+∠AFB=90° ∠DCF+∠DFC=90° ∠AFB=∠DFC
∴∠ABF=∠DCF
∵∠BAF=∠CAE=90° AB=AC
∴△ABF ≌△ACE
∴BF=CE
∴BF=2CD