证明：$\left(1\right)\because AF\cdot AB=AE^{2}$，

$\therefore \frac{AE}{AF}=\frac{AB}{AE}$，

$\because \angle EAB=\angle FAE$，

$\therefore \triangle EAF$∽$\triangle BAE$，

$\therefore \angle FEA=\angle B$；

$(2)\because \angle FEA=\angle B$，$\angle DAE=\angle CAB$，

$\therefore \angle DAF=\angle EAC,\triangle DAE$∽$\triangle CAB$，

$\therefore \angle D=\angle C$，$\frac{DE}{BC}=\frac{AD}{AC}$，

$\because \angle DAF=\angle EAC$，

$\therefore \triangle DAF$∽$\triangle CAE$，

$\therefore \frac{DA}{AC}=\frac{DF}{CE}$，

$\therefore \frac{DF}{AD}=\frac{CE}{AC}$.

证明：$\left(1\right)\because AF\cdot AB=AE^{2}$，

$\therefore \frac{AE}{AF}=\frac{AB}{AE}$，

$\because \angle EAB=\angle FAE$，

$\therefore \triangle EAF$∽$\triangle BAE$，

$\therefore \angle FEA=\angle B$；

$(2)\because \angle FEA=\angle B$，$\angle DAE=\angle CAB$，

$\therefore \angle DAF=\angle EAC,\triangle DAE$∽$\triangle CAB$，

$\therefore \angle D=\angle C$，$\frac{DE}{BC}=\frac{AD}{AC}$，

$\because \angle DAF=\angle EAC$，

$\therefore \triangle DAF$∽$\triangle CAE$，

$\therefore \frac{DA}{AC}=\frac{DF}{CE}$，

$\therefore \frac{DF}{AD}=\frac{CE}{AC}$.