Rt△ABC中,∠BAC=90°,AB=AC,BE平分∠ABC,AD⊥BC,CF⊥BE. 求证:1.AG=AE 2.BE
1个回答

(1) ∠AGE=∠BGD=∠ABG+∠BAG

∠AEG=∠EBC+∠ECB

AB=AC,∠BAC=90°,∴∠ACB=∠BAD=45°

而AE平分∠ABC,∴∠ABG=∠EBC

∴∠AGE=∠AEG,即AG=AE

(2) 设BE中点为M,连接AM,AF

∵∠BAC=∠BFC=90

∴B,A,F,C四点共圆,∴∠FAC=∠FBC=ABM

而AM=BM,∴∠ABM=∠BAM,∴∠FAC=∠MAB

同样∠ABM=∠ACF,∴△ABM∽△ACF

又AB=AC,∴△ABM≌△ACF,∴AM=AF

即AM=BM=AF=CF,∴BE=2BM=2CF

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