△ABC中,AD⊥BC于D,且CD=AB+BD,∠B的角平分线交AC于点E.求证:E恰好在BC的垂直平分线上.

Question

△ABC中,AD⊥BC于D,且CD=AB+BD,∠B的角平分线交AC于点E.求证:E恰好在BC的垂直平分线上.

Answer 1

在DC上取一点F,使DF=BD,连结AF
BD=DF,AD⊥BC
AB=AF,∠ABD=∠AFD
CD==CF+DF=AB+BD
CF=AB=AF
∠C=∠FAC
∠AFD=∠C+∠FAC=2∠C
BE是∠B的角平分线
∠ABD=2∠EBC
∠C=∠EBC
BE=CE
E恰好在BC的垂直平分线上.