Giải thích các bước giải:

a.Ta có : $DF//AM$

$\rightarrow \dfrac{DF}{AM}=\dfrac{DC}{MC}$

Mà $DE//AM\rightarrow \dfrac{DE}{AM}=\dfrac{BD}{BM}$

$\rightarrow \dfrac{DF}{AM}+\dfrac{DE}{AM}=\dfrac{DC}{MC}+\dfrac{BD}{BM}=\dfrac{DC}{MC}+\dfrac{BD}{MC}=\dfrac{BD+DC}{CM}=\dfrac{BC}{CM}=2$

$\rightarrow DE+DF=2AM$

b.Ta có :

$DE+DF=2AM\rightarrow DE+EN+DF-EN=2AM\rightarrow DN+DF-EN=2AM\rightarrow DF-EN=AM=DF-FN\rightarrow EN=FN$

$\rightarrow N$ là trung điểm EF

c.Ta có :

$(DM+MC)^2\ge 4DM.MC $

$\rightarrow(DM+MC)^4\ge 16DM^2.MC^2 $

$\rightarrow DC^4\ge 16DM^2.MC^2$

$\rightarrow (\dfrac{DC}{DM})^2.(\dfrac{DC}{MC})^2\ge 16$

$\rightarrow (\dfrac{DC}{NA})^2.(\dfrac{DC}{MC})^2\ge 16$

$\rightarrow \dfrac{S_{FDC}}{S_{FNA}}.\dfrac{S_{FDC}}{S_{AMC}}\ge 16$

$\rightarrow S^2_{FDC}\ge 16S_{AMC}.S_{FNA}$