Question

如图,在△ABC中,∠A=90°,D为△ABC内一点,且AB=AC=BD,∠ABD=30°,求证:

AD=CD

Answer 1

在△ABD内作正△ADE，联接BE
∴∠DAE＝60°
∵BA=BD AE=DE BE=BE
∴△ABE ≌△DBE
∴∠ABE=∠DBE
∵∠ABD＝30°
∴∠ABE=15°
∵ AB=BD
∴∠BAD=∠BDA=75°
∴∠BAE =∠BAD-∠DAE=15°
∴∠ABE=∠BAE
∴AE=BE
∵∠CAD =∠BAC-∠DAE-∠BAE=15°
∴∠BAE=∠CAD
∵AB=AC AE=AD
∴△ABE ≌△ACD
∴BE=CD
∵BE=AE=AD
∴AD=CD