解：如圖，連接??$OA、$????$OB、$????$OF，$??設(shè)??$OB$??交??$AF$??于點??$G$??
??$ ∵AB⊥CD，$????$CD$??是直徑
??$ ∴AE=BE=\frac 1 2AB=3$??
設(shè)??$⊙O$??的半徑為??$r，$??則??$OE=r-1，$????$OA=r$??
在??$Rt△OAE$??中，由勾股定理，得${3}^2+{(r-1)}^2={r}^2$??
解得，??$r=5$??
??$ ∵{\widehat{AB}}={\widehat{BF}}$??
??$ ∴∠AOB=∠FOB$??
??$ ∵AO=FO$??
??$ ∴OB⊥AF，$????$AF=2AG$??
設(shè)??$OG=t$??
∴在??$Rt△AGO$??中，??${AG}^2={5}^2-{t}^2$??
在??$Rt△AGB$??中，??${AG}^2={6}^2-{(5-t)}^2$??
??$ ∴{5}^2-{t}^2={6}^2-{(5-t)}^2$??
解得，??$t=\frac 7 5$??
??$ ∴AG=\sqrt {{5}^2-{(\frac 7 5)}^2}=\frac {24}5$??
??$ ∴AF=2AG=\frac {48}5$??
