如图，过$D$作$DF\bot AB$于$F$，

$\because \angle C=90^{\circ}$，$AD$是角平分线，
$\therefore DC=DF$，$\angle C=\angle DFB$，
又$\because DE=DB$，
$\therefore Rt\triangle CDE$≌$Rt\triangle FDB\left(HL\right)$，
$\therefore \angle B=\angle CED$，$\angle CDE=\angle FDB$，$CE=BF$，
又$\because \angle DEA+\angle DEC=180^{\circ}$，
$\therefore \angle DEA+\angle B=180^{\circ}$，故①正确；
$\because AD=AD$，$DC=DF$，
$\therefore Rt\triangle CDA$≌$Rt\triangle FDA\left(HL\right)$，
$\therefore AC=AF$，
$\therefore AB-AC=AB-AF=BF=CE$，故②正确；
$\because AC=AF$，
$\therefore AB+AE=\left(AF+FB\right)+\left(AC-CE\right)=AF+AC=2AC$，
$\therefore AC=\frac{1}{2}(AB+AE)$，
$\because CD\neq AE$，
$\therefore AC\neq \frac{1}{2}(AB+CD)$，故③错误；
$\because Rt\triangle CDE$≌$Rt\triangle FDB$，
$\therefore S_{\triangle CDE}=S_{\triangle FDB}$，
$\therefore S_{四边形ABDE}=S_{四边形ACDF}$，
又$\because \triangle ACD$≌$\triangle AFD$，
$\therefore S_{\triangle ACD}=S_{\triangle ADF}$，
$\therefore S_{\triangle ADC}=\frac{1}{2}S_{四边形ACDF}=\frac{1}{2}S_{四边形ABDE}$，故④正确；
$\therefore$一定正确的结论有①②④.
故答案为：①②④.

如图，过$D$作$DF\bot AB$于$F$，

$\because \angle C=90^{\circ}$，$AD$是角平分线，
$\therefore DC=DF$，$\angle C=\angle DFB$，
又$\because DE=DB$，
$\therefore Rt\triangle CDE$≌$Rt\triangle FDB\left(HL\right)$，
$\therefore \angle B=\angle CED$，$\angle CDE=\angle FDB$，$CE=BF$，
又$\because \angle DEA+\angle DEC=180^{\circ}$，
$\therefore \angle DEA+\angle B=180^{\circ}$，故①正确；
$\because AD=AD$，$DC=DF$，
$\therefore Rt\triangle CDA$≌$Rt\triangle FDA\left(HL\right)$，
$\therefore AC=AF$，
$\therefore AB-AC=AB-AF=BF=CE$，故②正确；
$\because AC=AF$，
$\therefore AB+AE=\left(AF+FB\right)+\left(AC-CE\right)=AF+AC=2AC$，
$\therefore AC=\frac{1}{2}(AB+AE)$，
$\because CD\neq AE$，
$\therefore AC\neq \frac{1}{2}(AB+CD)$，故③错误；
$\because Rt\triangle CDE$≌$Rt\triangle FDB$，
$\therefore S_{\triangle CDE}=S_{\triangle FDB}$，
$\therefore S_{四边形ABDE}=S_{四边形ACDF}$，
又$\because \triangle ACD$≌$\triangle AFD$，
$\therefore S_{\triangle ACD}=S_{\triangle ADF}$，
$\therefore S_{\triangle ADC}=\frac{1}{2}S_{四边形ACDF}=\frac{1}{2}S_{四边形ABDE}$，故④正确；
$\therefore$一定正确的结论有①②④.
故答案为：①②④.