$證明:過(guò)點(diǎn)E作EH⊥CB,交CB的延長(zhǎng)線于H$
$∵∠C=90°,BE⊥ AB$
$∴∠C=∠EBA=∠H=90°$
$∴∠ABC+∠A=90°,∠ABC+∠EBH=90°$
$∴∠A = ∠EBH$
$在 △ABC 和 △BEH 中$
$\begin{cases}∠C=∠H\\∠A=∠EBH\\AB=BE\end{cases}$
$∴△ABC≌△BEH(\mathrm {AAS})$
$∴EH=BC=BD$
$在△HEF和△BDF 中$
$\begin{cases}∠H=∠FBD\\∠EFH=∠DFB\\EH=BD\end{cases}$
$∴△HEF≌△BDF(\mathrm {AAS})$
$∴EF=DF$
$∴點(diǎn)F是ED的中點(diǎn)$