$ABCD$ is a cyclic quadrilateral whose diagonals $AC$ and $BD$ intersect at $E$. If $AB=CD$, prove that,

$AE=ED \text{and }EB=EC$

$ABCD$ is a cyclic quadrilateral whose diagonals $AC$ and $BD$ intersect at $E$. If $AB=CD$, prove that,

$AD\parallel BC$

In the adjoining figure, a chord $PQ$ of a circle with centre $O$ and radius $15 \text{cms}$ is bisected by a diameter $AB$. If $OM=9 \text{cm}$, find the length of $PQ.$

In the adjoining figure, a chord $PQ$ of a circle with centre $O$ and radius $15 \text{cms}$ is bisected by a diameter $AB$. If $OM=9 \text{cm}$, find the length of $AP.$

In the adjoining figure, a chord $PQ$ of a circle with centre $O$ and radius $15 \text{cms}$ is bisected by a diameter $AB$. If $OM=9 \text{cm}$, find the length of $BP.$

$ABC$ is an isosceles triangle inscribed in a circle. If $AB=AC=12\sqrt{5}\text{cm}$ and $BC=24\text{cm}$, find the radius of the circle?