If the perpendicular bisector of a chord $AB$ of a circle $PXAQBY$ intersects the circle at $P$ and $Q$, prove that  $\mathrm{arc} PXA\cong a\mathrm{rc} PYB$.

$ABCD$ is such a quadrilateral that $A$ is the centre of the circle passing through $B$, $C$ and $D$. Prove that $\angle CBD+\angle CDB=\frac{1}{2}\angle BAD$

$O$ is the circumcentre of the triangle $ABC$ and $D$ is the midpoint of the base $BC$. Prove that $\angle BOD=\angle A.$