?$(1)$?證明：∵?$∠ACB = 90°，$??$D$?是?$AB$?的中點(diǎn)，
∴?$DC = DB = DA，$?∴?$∠B = ∠DCB$?
∵?$?ABC≌?F DE，$?∴?$∠F DE = ∠B$?
∴?$∠F DE = ∠DCB，$?∴?$DG//BC$?
∴?$∠AG D = ∠ACB$?
∵?$∠ACB = 90°，$?∴?$∠AG D = 90°$?
?$(2)$?由?$(1)$?可得，?$∠AG D = 90°$?
∵?$DC = DA，$??$G D⊥AC，$?∴點(diǎn)?$G $?是?$AC$?的中點(diǎn)
∴?$CG = \frac 12\ \mathrm {A}C=\frac 12×8=4$?
∵點(diǎn)?$D$?是?$AB$?的中點(diǎn)，∴?$DG $?是?$?ABC$?的中位線
∴?$DG = \frac 12BC=\frac 12×6=3$?
∴?$S_{△DCG}=\frac 12CG·DG=\frac 12×4×3=6$?
∴圖?$1$?中重疊部分?$(?DCG)$?的面積為?$6$?
?$(3)$?連接?$BH$?
∵?$?ABC≌?F DE，$?∴?$∠ABC = ∠F DE$?
∵?$∠ACB = 90°，$??$DE⊥AB$?
∴?$∠A + ∠ABC = 90°，$??$∠A + ∠AHD = 90°$?
∴?$∠ABC = ∠AHD，$?∴?$∠A = ∠F DE，$?∴?$G D = GH$?
∵?$∠A + ∠AHD = 90°，$??$∠ADG + ∠F DE = 90°，$??$∠AHD = ∠F DE$?
∴?$∠A = ∠F DE，$?∴?$AG = G D，$?∴?$AG = CG$?
∴點(diǎn)?$G $?是?$AG $?的中點(diǎn)
∴?$S_{△DGH}=\frac 12S_{△ADH}$?
∵?$AB = \sqrt {AC^2+BC^2}，$??$AC = 8，$??$BC = 6$?
∴?$AB = 10$?
∴?$AD = \frac 12\ \mathrm {A}B=\frac 12×10=5$?
∴?$S_{△ADH}=\frac 12\ \mathrm {A}D·DH=\frac 12×5×\frac {15}4=\frac {75}8$?
∴?$S_{△DGH}=\frac 12 S_{△ADH}=\frac 12×\frac {75}8=\frac {75}{16}$?
∴重疊部分?$(?DGH)$?的面積為?$\frac {75}{16}$?
