證明：?$(1)∵D$?是?$\widehat{AC}$?的中點
?$∴\widehat{AD}=\widehat{CD}$?
?$∵AB⊥DH$?且?$AB$?是?$⊙O$?的直徑
?$∴\widehat{AD}=\widehat{AH}$?
?$∴\widehat{CD}=\widehat{AH}$?
?$∴∠ADH=∠CAD$?
?$∴AF=DF$?
?$(2)$?設?$AE=\sqrt{5}x，$?則?$AD=5x$?
?$∵ DE⊥AB，$?
?$∴∠AED=90°，$?
∴在?$Rt△AED$?中，?$DE= \sqrt{AD2-AE2}=2\sqrt{5}x.$?
?$∵AF=\frac {5}{2}，$??$AF=DF，$?
?$∴ DF=\frac {5}{2}.$?
∵在?$Rt△AEF $?中，?$AE2+EF2=AF2，$?
?$∴(\sqrt{5}x)2+(2\sqrt{5}x-\frac {5}{2})2=(\frac {5}{2})2，$?
解得?$x=\frac {2\sqrt{5}}{5}(x=0$?舍去)，
?$∴AE=\sqrt{5}x=2$?
