Question

如图，△ABC中，∠BAC=90°，AB=AC，BD是角平分线,CE⊥BD于E.求证BD=2CE

Answer 1

证明：△ABC中，∠BAC=90°，AB=AC，故∠ABC=∠ACB=45°
BD是角平分线,CE⊥BD于E. 得：∠ABD=∠EBC=∠ECD=22.5°
△ABD∽△ECD∽△EBC 故 CE/AB=DC/BD CE/BE=DC/BC AD/DE=BD/DC
又因：BC=√2AB BE=BD+DE AB=AD+DC 得：BD=2CE

Answer 2

证明:延长CE,交BA的延长线于F.
∠ABD=∠ACF(均为角F的余角);AB=AC;∠BAD=∠CAF=90°.
则⊿BAD≌⊿CAF,BD=CF.-----------------------------------(1)
∠CBE=∠FBE;BE=BE;∠BEC=∠BEF=90度.
则⊿BEC≌⊿BEF,CE=EF,CF=2CE.-----------------------(2)
所以,BD=2CE.