Question

AB.AC是圆O内相等的两弦,延长CA到D,使AD=AC,连DB并延长交圆O于E,连接BC,求证：∠EBC=90°.

Answer 1

∠ABC=∠BCA=X

∠ABD=∠ADB=Y

∠DAB=∠ABC+∠BCA=X+X=2X

∠BEC=∠DAB=2X

∠ECA=∠ADB=Y

∵∠BEC+∠ECA+∠ADB=180°

∴2X+Y+Y=180° 则 X+Y=90°

∴∠ABC+∠ABD=90°

又 ∠BEC=180°-∠ABC-∠ABD=180°-90°=90°

从而 ∠EBC=90°