解：猜想：$BF\perp AE。$
理由：$\because\angle ACB = 90^{\circ}$
$\therefore\angle ACE=\angle BCD = 90^{\circ}。$
在$Rt\triangle BDC$和$Rt\triangle AEC$中
$\begin{cases}BC = AC\\BD = AE\end{cases}，$
$\therefore Rt\triangle BDC\cong Rt\triangle AEC(HL)，$
$\therefore\angle CBD=\angle CAE。$
又$\because\angle CAE+\angle E = 90^{\circ}，$$\therefore\angle EBF+\angle E = 90^{\circ}，$
$\therefore\angle BFE = 90^{\circ}，$$\therefore BF\perp AE。$
