在三角形ABC中,∠C>∠B,AD⊥BC于D,AE平分∠BAC.求证:∠EAD=1|2(∠C-∠B).
问题描述:
在三角形ABC中,∠C>∠B,AD⊥BC于D,AE平分∠BAC.求证:∠EAD=1|2(∠C-∠B).
答
证明:延长BC到点F,使DF = DB
则:AD是BC的垂直平分线
∴AB =AF
∴∠B =∠F
∴ 2∠DAF = 2∠DAB = ∠BAF
∵∠ACB - ∠F = ∠FAC = ∠BAF - ∠BAC
= 2∠BAD - 2∠BAE =2(∠BAD - ∠BAE)= 2∠DAE
∴∠DAE = 1|2 (∠C - ∠B)