$\text{a) Xét ΔABC có:}$
$\text{$\widehat{ABC}$ + $\widehat{BAC}$ + $\widehat{ACB}$ = $180^{o}$ (đl tổng 3 góc trong Δ)}$
$\text{⇒ $\widehat{ABC}$ + $50^{o}$ + $\widehat{ACB}$ = $180^{o}$}$
$\text{⇒ $\widehat{ABC}$ + $\widehat{ACB}$ = $180^{o}$ - $50^{o}$}$
$\text{⇒ $\widehat{ABC}$ + $\widehat{ACB}$ = $130^{o}$}$
$\text{mà $\widehat{ABC}$ = $\widehat{ACB}$ (ΔABC cân tại A)}$
$\text{⇒ $\widehat{ABC}$ = $\widehat{ACB}$ = $\dfrac{130^o}{2}$ = $65^{o}$}$
$\text{b) Có: MD ⊥ BC (gt); NE ⊥ BC (gt)}$
$\text{⇒ MD // NE (⊥ → //)}$
$\text{Có: $\widehat{ABC}$ = $\widehat{ACB}$ (cmt)}$
$\text{mà $\widehat{ACB}$ = $\widehat{ECN}$ (đối đỉnh)}$
$\text{⇒ $\widehat{ABC}$ = $\widehat{ECN}$}$
$\text{Xét ΔMBD và ΔNCE có:}$
$\text{$\widehat{ABC}$ = $\widehat{ECN}$ (cmt)}$
$\text{BD = CE (gt)}$
$\text{$\widehat{MDB}$ = $\widehat{NEC}$ = $90^{o}$}$
$\text{⇒ ΔMBD = ΔNCE (g.c.g)}$
$\text{⇒ MD = NE (2 cạnh t/ứ)}$
$\text{c) Xét ΔMDI và ΔNEI có:}$
$\text{$\widehat{MDI}$ = $\widehat{NEI}$ = $90^{o}$}$
$\text{MD = NE (cmt)}$
$\text{$\widehat{MID}$ = $\widehat{NIE}$ (đối đỉnh)}$
$\text{⇒ ΔMDI = ΔNEI (ch-gn)}$
$\text{⇒ DI = EI (2 cạnh t/ứ)}$
$\text{⇒ I là trung điểm DE}$
