證明：?$(1) ∵ OA⊥BD，$?
?$∴ ∠AOB=∠AOD=90°，$?
?$∴ \widehat{AB}=\widehat{AD}$?
?$∴ ∠ACB=∠ACD，$?
?$∴CA$?平分?$∠BCD$?
?$(2)$?如圖，延長?$AE$?交?$BC$?于點?$M，$?延長?$CE$?交?$AB$?于點?$N.$?
?$∵ AE⊥BC，$??$CE⊥AB，$?
?$∴ ∠AMB=∠CNB=90°.$?
?$∵ BD$?是?$⊙O$?的直徑，
?$∴ ∠BAD=∠BCD=90°，$?
?$∴ ∠BAD=∠CNB，$??$∠BCD=∠AMB，$?
?$∴ AD//NC，$??$CD//AM，$?
∴ 四邊形?$AECD$?是平行四邊形，
?$∴ CD=AE=3，$?
∴ 在?$Rt△BCD$?中?$，BC= \sqrt{BD-CD2} = \sqrt{(3\sqrt{3})2-32} =3 \sqrt{2}$?
