?$(1)$?證明：∵?$AC⊥BD，$?且?$∠CAD=45°$?
∴?$AC=CD$?
在?$Rt△ABC$?和?$Rt△DEC$?中
?$\begin {cases}{AC=DC}\\{AB=DE}\end {cases}$?
∴?$Rt△ABC≌Rt△DEC(\mathrm {HL})$?
?$(2)$?∵?$△ ABC≌△DEC，$?∴?$∠BAC=∠EDC$?
∵?$∠EDC+∠CED=90°，$??$∠CED=∠AEF$?
∴?$∠AEF+∠BAC=90°，$?∴?$∠AFE=90°，$?∴?$DF⊥AB$?
?$(3)$?∵?$S_{△BCE}+S_{△ACD}=S_{△ABD}-S_{△ABE}$?
∴?$\frac 12a^2+\frac 12b^2=\frac 12·c·DF-\frac 12·c·EF$?
?$=\frac 12·c·(DF-EF)=\frac 12·c·DE=\frac 12c^2$?
∴?$a^2+b^2=c^2$?
