1、当$D$，$E$，$F$共线时，$AF$最小，如图所示，$\because AB=AC$，$AB=DF$，$\therefore AC=DF$，又$\angle FDC=\angle ACD=45^{\circ}$，$\because DO=OC$，$\therefore OA=OF$，$\because \angle AOF=90^{\circ}$，$\therefore AF=\sqrt{2}AO$，当$AO$有最小值时，$AF$最小，即当$O$在$AC$上时，此时$D$，$E$，$F$共线，$\because CD=\sqrt{2}$，$\therefore CO=1$，$\because AC=\sqrt{5}$，$\therefore AO=\sqrt{5}-1$，$\therefore AF=\sqrt{2}AO=\sqrt{2}(\sqrt{5}-1)=\sqrt{10}-\sqrt{2}$.

2、当$D$，$E$，$F$共线时，$AF$最小，如图所示，$\because AB=AC$，$AB=DF$，$\therefore AC=DF$，又$\angle FDC=\angle ACD=45^{\circ}$，$\because DO=OC$，$\therefore OA=OF$，$\because \angle AOF=90^{\circ}$，$\therefore AF=\sqrt{2}AO$，当$AO$有最小值时，$AF$最小，即当$O$在$AC$上时，此时$D$，$E$，$F$共线，$\because CD=\sqrt{2}$，$\therefore CO=1$，$\because AC=\sqrt{5}$，$\therefore AO=\sqrt{5}-1$，$\therefore AF=\sqrt{2}AO=\sqrt{2}(\sqrt{5}-1)=\sqrt{10}-\sqrt{2}$.

【#如图 $\triangle ABC$和$\triangle DCE$都为等腰直角三角形 $\angle BAC=\angle DCE=90^{\circ}$ 连#】到此分享完毕，希望对大家有所帮助。