Chứng minh tứ giác $IODB$ nội tiếp như sau$\widehat{BIO}=180-\widehat{IBC}-\widehat{ICB}=180-\frac{\widehat{B}}{2}-\frac{\widehat{C}}{2}$
$\widehat{BDO}=90-\widehat{IBC}=90-\frac{\widehat{B}}{2}$ Do $OD$ vuông góc $BI$
$\widehat{BDO}+\widehat{BIO}=270-\frac{\widehat{B}}{2}-\frac{\widehat{B}}{2}-\frac{\widehat{C}}{2}=180$ Do $\widehat{A}=\widehat{B}\Rightarrow 2\widehat{B}+\widehat{C}=180$
Vậy $IODB$ nội tiếp
$\widehat{OID}=\widehat{OBD}$
$\widehat{OBD}=\widehat{OCD}$ Do $\triangle OBC$ cân
$\widehat{OCD}=\widehat{OCA}$
$\Rightarrow \widehat{OID}=\widehat{OCA}$
$\Rightarrow ID//CA$