Question

如图所示,已知三角形abc中,ac等于bc,角acb等于90度,bd为角abc的平分线,ad垂直于bd于d.求证be等于2ad

Answer 1

延长AD和BC交于点F
∠DAE+∠AED=90,∠CBE+∠CEB=90
∠DAE+∠AED=∠CBE+∠CEB=90
因∠AED=∠CEB
所以，∠DAE＝∠CBE
∠ACF＝∠BCE=90,AC=BC,∠DAE＝∠CBE
RT△ACF≌RT△BCE(ASA)
所以，BE=AF,CF=CE
DB是∠ABC平分线，∠ABD=∠DBF
∠ABD=∠DBF,DB=BD∠ADB=∠BDF=90
RT△ABD≌RT△BDF(ASA)
所以，AD=DF
AF=AD+DF=2AD
AF=2AD
又BE=AF
即有，BE=2AD