Cho $\triangle ABC$ cân tại B có $\widehat{BAC}=53^o$, BN là tia phân giác của góc B.
a) Tính số đo của góc ABC
b) Chứng minhg: $\triangle BAN = \triangle BCN$
c) Kẻ $AE \perp BC$ ($E\in BC$), $CI \perp AB$ ($I \in AB$). Chứng minh: $\triangle CEA = \triangle AIC$
d) Chứng minh: $AC // IE$
e) Gọi S là giao điểm của AE và CI. Chứng minh: B,S,N thẳng hàng
Bài giải:
![](data:image/webp;base64,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)
a) Tính số đo góc ABC:
Do $\triangle ABC$ cân tại B nên:
$\widehat{BCA}=\widehat{BAC}=53^o$
Ta có:$\widehat{ABC}+\widehat{BCA}+\widehat{BAC} = 180^o$
$⇔ \widehat{ABC} + 53^o+53^o=180^o$
$⇔ \widehat{ABC} + 106^o=180^o$
$⇔ \widehat{ABC} = 180^o - 106^o$
$⇔ \widehat{ABC} = 74^o$
Vậy $\widehat{ABC} = 74^o$
b) Chứng minhg: $\triangle BAN = \triangle BCN$
Ta có:
$\begin{cases}
BA = BC \text{ (} \triangle ABC \text{ cân tại B)}\\
BN \text{ cạnh chung}\\
\widehat{ABN}=\widehat{CBN} \text{( BN là tia phân giác)}
\end{cases}$
$\Rightarrow \triangle BAN = \triangle BCN$ (c-g-c)
c) Chứng minh: $\triangle CEA = \triangle AIC$
Xét hai tam giác vuông CEA và AIC:
$\begin{cases}
\text{Cạnh huyền } AC \text{ chung}\\
\widehat{ECA}=\widehat{IAC} \text{ (} \triangle ABC \text{ cân tại B)}
\end{cases}$
$\Rightarrow \triangle CEA = \triangle AIC$ (cạnh huyền-góc nhọn)
d) Chứng minh: $AC // IE$
Từ $\triangle CEA = \triangle AIC$
$\Rightarrow CE = AI$
$\Rightarrow BE = BI$
$\Rightarrow \triangle BEI$ cân tại B
$\Rightarrow \widehat{BEI} = \widehat{BCA} = \frac{180^o-\widehat{EBI}}{2}$ (ở vị trí đồng vị)
$\Rightarrow AC //IE$
e) Chứng minh B,S,N thẳng hàng
Xét $\triangle ABC$:
$\begin{cases}
CI \perp AB\\
AE \perp BC\\
S = CE \cap CI
\end{cases}$
$\Rightarrow S$ là trực tâm của $\triangle ABC$
Ngoài ra $\triangle ABC$ cân tại B nên đường phân giác BN cũng là đường cao
$\Rightarrow S \in BN$
hay B,S,N thẳng hàng