In a circle with centre $P$,$AB$ and $CD$ are equal chords. If $\angle APB = 80^{\circ},$ then $\angle CPD =$ .......... (in $^{\circ}$)

If $ABC$ is an equilateral triangle inscribed in a circle and $P$ is any point on the minor arc $BC$ which does not coincide with $B$ or $C$,prove that $PA$ is the angle bisector of $\angle BPC$.

The chord which passes through the centre of the circle is called a .......... of the circle.

If $AB = 12 \, cm$,$BC = 16 \, cm$ and $AB$ is perpendicular to $BC$,then the radius of the circle passing through the points $A, B$ and $C$ is (in $cm$)

In the figure,$AOC$ is a diameter of the circle and $\operatorname{arc} AXB = \frac{1}{2} \operatorname{arc} BYC$. Find $\angle BOC$.

$AB$ and $AC$ are two equal chords of a circle. Prove that the bisector of the angle $BAC$ passes through the centre of the circle.