证明：$\left(1\right)\because \angle C=90^{\circ}$，
$\therefore DC\bot AC$，
$\because AD$是$\angle BAC$的平分线，$DE\bot AB$，
$\therefore DE=DC$，
在$Rt\triangle CDF$与$Rt\triangle EDB$中，
$\left\{\begin{array}{l}{DC=DE}\\{DF=DB}\end{array}\right.$，
$\therefore Rt\triangle CDF$≌$Rt\triangle EDB\left(HL\right)$，
$\therefore CF=EB$；
$(2)$在$Rt\triangle ACD$和$Rt\triangle AED$中，
$\left\{\begin{array}{l}{DC=DE}\\{AD=AD}\end{array}\right.$，
$\therefore Rt\triangle ACD$≌$Rt\triangle AED\left(HL\right)$，
$\therefore AC=AE$，
$\because CF=BE$，
$\therefore AB=AC+EB=AF+2EB$.

证明：$\left(1\right)\because \angle C=90^{\circ}$，
$\therefore DC\bot AC$，
$\because AD$是$\angle BAC$的平分线，$DE\bot AB$，
$\therefore DE=DC$，
在$Rt\triangle CDF$与$Rt\triangle EDB$中，
$\left\{\begin{array}{l}{DC=DE}\\{DF=DB}\end{array}\right.$，
$\therefore Rt\triangle CDF$≌$Rt\triangle EDB\left(HL\right)$，
$\therefore CF=EB$；
$(2)$在$Rt\triangle ACD$和$Rt\triangle AED$中，
$\left\{\begin{array}{l}{DC=DE}\\{AD=AD}\end{array}\right.$，
$\therefore Rt\triangle ACD$≌$Rt\triangle AED\left(HL\right)$，
$\therefore AC=AE$，
$\because CF=BE$，
$\therefore AB=AC+EB=AF+2EB$.