如图,尺规作图痕迹与$\triangle ABC$的边$BC$、$AB$分别交于点$D$、$E$,过点$D$作$DF\bot AB$于点$F$,在$AC$上取一点$G$,使$DE=DG$,若$\triangle ADG$的面积为$52$,$\triangle AED$的面积为$38$,则$\triangle DEF$的面积为____.