E là trung điểm $BC\Rightarrow EB=EC=a$

$\Rightarrow AE$ là trung tuyến $\Delta ABC, DE$ là trung tuyến $\Delta BCD $

Mà $\Delta ABC$ cân tại $A(AB=AC); \Delta BCD$ cân tại $D(BD=CD)$

$\Rightarrow AE \perp BC,DE \perp BC\\ BC=(ABC) \subset (BCD)\\ \Rightarrow ((ABC);(BCD))=(AE,DE)=\widehat{AED}=45^o\\ \Delta AEC; \widehat{E}=90^o\\ \Rightarrow AE=\sqrt{AC^2-EC^2}=a\\ AE \perp BC,DE \perp BC\\ \Rightarrow BC \perp (ADE)$

Trong $(ADE)$, kẻ $AH \perp ED$

Mà $AH \perp BC(BC \perp (ADE);AH \subset (ADE))$

$\Rightarrow AH \perp (BCD)\\ \Delta AEH, \widehat{AEH}=45^o,\widehat{AHE}=90^o\\ \Rightarrow AH=EH=AE\cos(45^o)=\dfrac{\sqrt{2}}{2}a\\ \Delta BED,\widehat{BED}=90^o\\ \Rightarrow ED=\sqrt{BD^2-EB^2}=a\sqrt{2}\\ \Rightarrow HD=ED-EH=\dfrac{\sqrt{2}}{2}a\\ \Delta AHD, AH \perp HD\\ \Rightarrow AD=\sqrt{AH^2+HD^2}=a\\ \Delta ABC, AB^2+AC^2=BC^2 \Rightarrow AB \perp AC\\ \Delta ABD, AB^2+AD^2=BC^2 \Rightarrow AB \perp AD\\ \Rightarrow AB \perp (ACD)\\ \Rightarrow d(B,(ACD))=AB=a\sqrt{2}$