$\hspace{0.65cm}\ \widehat{BAE}$ là góc nội tiếp $(O)$ chắn $\mathop{BE}\limits^{\displaystyle\frown}$

$\Rightarrow\ \widehat{BAE}=\dfrac{1}{2}sđ\mathop{BE}\limits^{\displaystyle\frown}\ \ \ \ \ \ \ \ (1)$

$\hspace{0.65cm}\ \widehat{CAE}$ là góc nội tiếp $(O)$ chắn $\mathop{CE}\limits^{\displaystyle\frown}$

$\Rightarrow\ \widehat{CAE}=\dfrac{1}{2}sđ\mathop{CE}\limits^{\displaystyle\frown}\ \ \ \ \ \ \ \ (2)$

$\hspace{0.65cm}\ AE$ là phân giác $\widehat{BAC}\ \ (gt)$

$\Rightarrow\ \widehat{BAE}=\widehat{CAE}\ \ \ \ \ \ \ \ (3)$

$\hspace{0.65cm}\ $Từ $(1),(2),(3)\ \Rightarrow\ sđ\mathop{BE}\limits^{\displaystyle\frown}=sđ\mathop{CE}\limits^{\displaystyle\frown}$

$\Rightarrow\ BE=CE$

$\Rightarrow\ E\in$ đường trung trực $BC\ \ \ \ \ \ \ \ (4)$

$\hspace{0.65cm}\ OB=OC=R$

$\Rightarrow\ O\in$ đường trung trực $BC\ \ \ \ \ \ \ \ (5)$

$\hspace{0.65cm}\ $Từ $(4),(5)\ \Rightarrow\ OE$ là đường trung trực $BC$

$\Rightarrow\ OE\bot BC\ \ (đpcm)$

$\\$

$\hspace{0.65cm}\ OA=OE=R$

$\Rightarrow\ \triangle OAE$ cân tại $O$

$\Rightarrow\ \widehat{OAE}=\widehat{OEA}\ \ \ \ \ \ \ \ (6)$

$\hspace{0.65cm}\ AH$ là đường cao của $\triangle ABC$

$\Rightarrow\ AH\bot BC$

$\hspace{0.65cm}\ $Mà $OE\bot BC\ (cmt)$

$\Rightarrow\ AH//OE$

$\Rightarrow\ \widehat{HAE}=\widehat{OEA}\ \ \ \ \ \ \ \ (7)$

$\hspace{0.65cm}\ $Từ $(4),(5)\ \Rightarrow\ \widehat{HAE}=\widehat{OAE}$

$\Rightarrow\ AE$ là phân giác $\widehat{HAO}\ \ (đpcm)$