證明：?$(1)∵C$?是?$\widehat{BD}$?的中點(diǎn)，
?$∴\widehat{CD}=\widehat{BC}.$?
?$∵AB$?是?$⊙O$?的直徑，且?$CF⊥AB，$?
?$∴\widehat{BC}=\widehat{BF}，$?
?$∴\widehat{CD}=\widehat{BF}，$?
?$∴CD=BF.$?
∵在?$△BFG $?和?$△CDG $?中，?$∠F=∠CDG，$??$∠FGB=∠DGC，$??$BF=CD，$?
?$∴△BFG≌△CDG.$?
?$(2)$?如圖，過(guò)?$C$?作?$CH⊥AD$?于?$H，$?連接?$AC、$??$BC，$?
?$∵\(yùn)widehat{CD}=\widehat{BC}，$?
?$∴∠HAC=∠BAC.$?
?$∵CE⊥AB，$?
?$∴CH=CE.$?
?$∵AC=AC，$?
?$∴Rt△AHC≌Rt△AEC，$?
?$∴AE=AH.$?
?$∵CH=CE，$??$CD=CB，$?
?$∴Rt△CDH≌Rt△CBE，$?
?$∴DH=BE=2，$?
?$∴AE=AH=2+2=4，$?
?$∴AB=4+2=6.$?
?$∵AB$?是?$⊙O$?的直徑，
?$∴∠ACB=90°，$?
?$∴∠ACB=∠BEC=90°.$?
?$∵∠EBC=∠ABC，$?
?$∴△BEC∽△BCA，$?
?$∴\frac {BC}{AB}=\frac {BE}{BC}，$?
?$∴BC^2=AB·BE=6×2=12，$?
?$∴BF=BC=2\sqrt{3}.$?
