證明：?$(1)$?如圖，連接?$OC。$?
∵?$ C$?是?$\overset {\frown }{ACB}$?的中點，
∴?$ \overset {\frown }{AC}=\overset {\frown }{BC}，$?
根據(jù)在同圓或等圓中，相等的弧所對的圓心角相等，可得?$∠COD = ∠COE。$?
∵?$ OA = OB，$??$AD = BE，$?
∴?$ OA - AD = OB - BE，$?即?$OD = OE。$?
又∵?$ OC = OC，$?
?$ $?在?$△COD$?和?$△COE$?中，
?$ \begin {cases}OD = OE\\∠COD=∠COE\\OC = OC\end {cases}$?
?$ $?根據(jù)?$SAS($?邊角邊?$)$?判定定理，可得?$△COD≌△COE，$?
∴?$ CD = CE。$?
?$ (2)$?證明：
如圖，連接?$OM、$??$ON。$?
∵?$ △COD≌△COE，$?
∴?$ ∠CDO = ∠CEO，$??$∠OCD = ∠OCE。$?
∵?$ OC = OM = ON，$?
∴?$ ∠OCM = ∠M，$??$∠OCN = ∠N，$?
∴?$ ∠M = ∠N。$?
∵?$ ∠CDO = ∠M + ∠MOD，$??$∠CEO = ∠N + ∠NOE，$?
∴?$ ∠MOD = ∠NOE，$?
根據(jù)在同圓或等圓中，相等的圓心角所對的弧相等，可得?$\overset {\frown }{AM}=\overset {\frown }{BN}。$?
