AD平分∠BAC交BC于D,∠ABC、∠ACB的平分线交AD于O,过O点作OE⊥BC于E.求证：∠BOD=∠EOC.

题目

AD平分∠BAC交BC于D,∠ABC、∠ACB的平分线交AD于O,过O点作OE⊥BC于E.求证：∠BOD=∠EOC.

答案

证明:
∵∠BOD=∠ABO+∠BAO=∠BAC/2+∠ABC/2=(180度-∠ACB)/2=90度-∠ACB/2
=90度-∠OCB
∠EOC=90度-∠OCB
∴∠BOD=∠EOC