$證明:延長AF至點G,使FG=AF,連接BG$
$∵F為BE的中點,∴EF=BF$
$在△AFE和△GFB中$
$\begin{cases}{ AF=GF }\ \\ { ∠AFE=∠GFB } \\{ EF=BF} \end{cases}$
$∴△AFE≌△GFB(SAS),∴∠EAF=∠G,AE=BG\ $
$∴AE//BG,∴∠GBA+∠BAE=180°\ $
$∵∠BAC+∠EAD=180°,∴∠DAC+∠BAE=180°\ $
$∴∠GBA=∠DAC$
$∵AD=AE,∴BG=AD$
$在△GBA和△DAC中$
$\begin{cases}{ AB=CA }\ \\ { ∠GBA=∠DAC } \\{ BG=AD} \end{cases}$
$∴△GBA≌△DAC(SAS),∴AG=CD\ $
$∵AG=2AF,∴CD=2AF $