Question

如图，三角形ABC中，AB=AC,AD是∠BAC的角平分线，AE是∠BAC的外角的平分线，BE⊥AE.

求证DA⊥AE.

Answer 1

∵AD是∠BAC的平分线，AE是∠EAF的平分线
∴∠BAD=1/2∠BAC，∠BAE=1/2∠EAF
∴∠DAE =∠BAD+∠BAE = 1/2∠BAC+1/2∠EAF = 1/2∠CAF = 90°
∴DA⊥AE.

Answer 2

因为∠DAB=∠DAC ∠BAE=∠FAE
所以∠DAB+∠BAE=∠DAC+∠FAE 即∠DAE=∠DAC+∠FAE
∠DAE+∠DAC+∠FAE=180°所以2∠DAE=180°所以∠DAE=90°所以DA⊥AE