(1)證明: ?$∵AE\perp AD，EF\perp AB，$?
?$∴∠DAB+∠EAB=90°，∠E+∠EAB=90°，$?
?$∴∠DAB=∠E，$?
?$∵∠C=90°，AC=BC，$?
?$∴∠CAB=∠B=45°$?
?$∵EF\perp AB，$?
?$∴∠AFE=∠CAB=∠B=45°，$?
在?$\triangle EAF$?和?$\triangle ADB$?中,
?$\{\begin {array}{l} ∠E=∠DAB\\∠AFE=∠B，\\AE=AD\end {array}.$?
?$∴\triangle EAF\cong \triangle ADB(\mathrm {AAS})；$?
(2)證明: 在?$GE$?上截取?$GH=GF$?,連接?$AH$?.
?$∵GH=GF=AG，EF\perp AB，$?
?$∴AF=AH，∠GAH=∠AHG=45°，$?
?$∴∠FAH=∠DAE=90°，$?
?$∴∠FAD=∠HAE，$?
在?$\triangle FAD$?和?$\triangle HAE$?中,
?$\{\begin {array}{l}\ \mathrm {A}F=AH\\∠FAD=∠HAE，\\AD=AE\end {array}.$?
?$∴\triangle FAD\cong \triangle HAE(\mathrm {SAS})，$?
?$∴EH=DF，$?
?$∴EF=FH+EH=2AG+DF；$?
