$F$ là giao $AC,BD\Rightarrow F$ là trung điểm $AC,BD$
$E$ là trung điểm $SA$
$\Rightarrow EF$ là đường trung bình $\Delta SAC$
$\Rightarrow EF //SC\\ \Rightarrow (BED) //SC\\ BD \subset (BED)\\ \Rightarrow d(SC;BD)=d(SC;(BED))=d(C;(BED))=d(A;(BED))$
Kẻ $AG \perp BD$
$EA \perp BD(SA \perp (ABCD))\\ \Rightarrow BD \perp (EAG)\\ \Rightarrow (BED) \perp (EAG)$
Kẻ $AH \perp EG$
$EG=(BED) \cap (EAG)\\ \Rightarrow AH \perp (BED)\\ \Delta ABD,AB \perp AD,AG \perp BD\\ \Rightarrow AG=\sqrt{\dfrac{1}{\dfrac{1}{AB^2}+\dfrac{1}{AD^2}}}=\dfrac{2\sqrt{5}}{5}\\ \Delta EAG,EA \perp AG(SA \perp (ABCD)) ,AH \perp EG\\ \Rightarrow AH=\sqrt{\dfrac{1}{\dfrac{1}{EA^2}+\dfrac{1}{AG^2}}}=\dfrac{2\sqrt{21}}{21}$
