$ABCD$ is a square. $M$ is the mid-point of $AB$ and $PQ\perp CM$ meets $AD$ at $P$ and $CB$ produced at $Q$.

Prove that $PA=BQ$

$ABCD$ is a square. $M$ is the mid-point of $AB$ and $PQ\perp CM$ meets $AD$ at $P$ and $CB$ produced at $Q$.

Prove that $CP=AB+PA$

In the figure, $\mathrm{OA}=\mathrm{OB}, \mathrm{OC}=\mathrm{OD},\angle \mathrm{AOB}=\angle \mathrm{COD}$. Prove that $\mathrm{AC}=\mathrm{BD}$.