a) Xét $ΔABC$ có:
$AD = DC = \dfrac{1}{2}AC \quad (gt)$
$AE = EB = \dfrac{1}{2}AB \quad (gt)$
$\Rightarrow DE$ là đường trung bình
$\Rightarrow DE//BC;\, DE = \dfrac{1}{2}BC = \dfrac{1}{2}.4 = 2\, cm$
b) Xét hình thang $BCDE \quad (DE//BC)$ có:
$BM = ME = \dfrac{1}{2}BE \quad (gt)$
$CN = ND = \dfrac{1}{2}CD\quad (gt)$
$\Rightarrow MN$ là đường trung bình
$\Rightarrow MN//DE//BC$
Xét $ΔEDC$ có:
$NQ//ED \quad (MN//ED)$
$CN = ND = \dfrac{1}{2}CD$
$\Rightarrow EQ = QC = \dfrac{1}{2}EC$
$\Rightarrow NQ$ là đường trung bình
$\Rightarrow NQ = \dfrac{1}{2}ED$
Xét $ΔDBC$ có:
$NP//BC\quad (MN//BC)$
$CN = ND = \dfrac{1}{2}CD$
$\Rightarrow BP = PD = \dfrac{1}{2}BD$
$\Rightarrow NP$ là đường trung bình
$\Rightarrow NP = \dfrac{1}{2}BC = ED$
$\Rightarrow NP = 2NQ$
$\Rightarrow NP - NQ = NQ$
$\Rightarrow PQ = NQ$
