Question

In a quadrilateral $ABCD,AB=AD$ and $CB=CD$. Then prove that $AC$ is perpendicular bisector of $BD$.

Answer 1

In $△AOB$ and $△AOD$

$AB=AD$ (given)

$\angle BAD=\angle DAO$ (AC bisects $\angle A$)

$\therefore AO=AO$ [common]

$\Rightarrow △AOB\cong △AOD$ [SAS]

$DO=BO$.....(i) [CPCT]

$\angle AOD=\angle AOB$

$\angle AOD+\angle AOB={180}^o$

$\Rightarrow \angle AOD=\angle AOB={90}^o$.....(ii)

From (i) and (ii)

$AC$ is perpendicular bisector of $BD$