證明：?$(1)$?因為?$FA = FE，$?
所以?$∠FAE = ∠AEF。$?
因為?$\overset {\frown }{BF}=\overset {\frown }{BF}，$?
所以?$∠FAE = ∠BCE。$?
因為?$∠AEF = ∠CEB，$?
所以?$∠CEB = ∠BCE。$?
因為?$CE$?平分?$∠ACD，$?
所以?$∠ACE = ∠DCE。$?
因為?$AB$?是?$\odot O$?的直徑，
所以?$∠ACB = 90°，$?即?$∠BCE+∠ACE = 90°，$?
所以?$∠CEB+∠DCE = 90°。$?
因為?$\triangle CDE$?的內角和為?$180°，$?
所以?$∠CDE = 90°，$?
所以?$CD\perp AB。$?
?$(2)$?由?$(1)$?知，?$∠BEC = ∠BCE，$?
所以?$BE = BC。$?
因為?$OM = OE = 1，$?
所以?$ME = OM + OE = 2。$?
因為?$AF = EF，$??$FM\perp AB，$?
所以?$MA = ME = 2，$?
所以?$AE = 4，$?
所以?$OA = OB = AE - OE = 3，$?
所以?$BC = BE = OB - OE = 2，$??$AB = OA + OB = 6。$?
在?$Rt\triangle ABC$?中，?$AC=\sqrt {AB^2-BC^2}=\sqrt {6^2-2^2} = 4\sqrt {2}。$?
