证明：如图，连接$AD$，
$\because$直线$MN$是线段$AB$的垂直平分线，
$\therefore AD=BD$，
$\therefore \angle DAB=\angle B$，
又$\because \angle B=30^{\circ}$，
$\therefore \angle DAB=30^{\circ}$，
又$\because AB=AC$，$\angle B=30^{\circ}$，
$\therefore \angle B=\angle C=30^{\circ}$，$\angle BAC=120^{\circ}$，
$\therefore \angle DAC=90^{\circ}$，
又$\because \angle C=30^{\circ}$，
$\therefore CD=2AD$，
又$\because AD=BD$，
$\therefore CD=2BD$.

证明：如图，连接$AD$，
$\because$直线$MN$是线段$AB$的垂直平分线，
$\therefore AD=BD$，
$\therefore \angle DAB=\angle B$，
又$\because \angle B=30^{\circ}$，
$\therefore \angle DAB=30^{\circ}$，
又$\because AB=AC$，$\angle B=30^{\circ}$，
$\therefore \angle B=\angle C=30^{\circ}$，$\angle BAC=120^{\circ}$，
$\therefore \angle DAC=90^{\circ}$，
又$\because \angle C=30^{\circ}$，
$\therefore CD=2AD$，
又$\because AD=BD$，
$\therefore CD=2BD$.