证明：$(1)$在$ \triangle AOB $和$ \triangle COD $中
$\begin {cases}{∠A = ∠C}\\{OA = OC}\\{∠AOB = ∠COD}\end {cases}$
∴$\triangle AOB ≌ \triangle COD (AS A)，$∴$ OB = OD $
$(2)$∵$OB = OD，$∴点$ O $在$ BD $的垂直平分线上
又∵$BE = DE，$∴点$ E $在$ BD $的垂直平分线上
∴$ OE $垂直平分$ BD $
