\(a,\left\{{}\begin{matrix}AE=EB\\AD=DC\end{matrix}\right.\Rightarrow ED\) là đtb tam giác ABC
\(\Rightarrow ED=\dfrac{1}{2}BC;ED//BC\Rightarrow BEDC\) là hthang
\(b,\left\{{}\begin{matrix}EM=MB\\DN=NC\end{matrix}\right.\Rightarrow MN\) là đtb hthang BEDC
\(\Rightarrow MN//DE//BC;MN=\dfrac{DE+BC}{2}\)
Mà \(EM=MB\Rightarrow BI=ID\Rightarrow MI\) là đtb tam giác BED
\(\Rightarrow MI=\dfrac{1}{2}DE=0,5DE=\dfrac{1}{2}\cdot\dfrac{1}{2}BC=\dfrac{1}{4}BC=0,25BC\)
\(c,\) \(\left\{{}\begin{matrix}NK//ED\\DN=NC\end{matrix}\right.\Rightarrow EK=KC\Rightarrow KN\) là đtb tam giác EDC
\(\Rightarrow KN=\dfrac{1}{2}ED=MI\left(1\right)\)
\(IK=MN-MI-KN=\dfrac{ED+BC}{2}-\dfrac{ED}{2}-\dfrac{ED}{2}\\ =\dfrac{BC-DE}{2}=\dfrac{2DE-DE}{2}=\dfrac{DE}{2}=MI=KN\left(2\right)\)
\(\left(1\right)\left(2\right)\Rightarrow MI=IK=KN\)
\(d,IN=NK+KI=\dfrac{1}{2}DE+\dfrac{1}{2}DE=DE;IN//DE\left(MN//DE\right)\)
\(\Rightarrow EDNI\) là hbh nên \(EI=ND\)
