$證明:如答圖①,取AC的中點F,連接EF$
$則AF= \frac{1}{2}AC$
$∵Rt△ABC中,∠ACB=30°$
$∴AB=\frac{1}{2}AC,∠BAC=60°,∴AB=AF\ $
$∵△ADE是等邊三角形,∴∠DAE=60°,AD=AE\ $
$∴∠BAD+∠DAC=∠DAC+∠CAE=60°\ $
$∴∠BAD=∠FAE$
$在△ABD和△AFE中$
${{\begin{cases} {{AB=AF}} \\ {∠BAD=∠FAE} \\ {AD=AE} \end{cases}}}$
$∴△ABD≌△AFE(SAS),∴∠AFE=∠B=90°$
$∴EF垂直平分AC,∴AE=CE,∴DE=CE $