The line segment joining the centre and any point on the circle is called a .......... of the circle.

In an isosceles triangle $ABC$ with $AB = AC$,a circle passing through $B$ and $C$ intersects the sides $AB$ and $AC$ at $X$ and $Y$ respectively. Prove that $XY \parallel BC$.

Two circles of radii $8 \ cm$ with centers $P$ and $Q$ are given. The chord $AB$ of the circle with center $P$ and the chord $CD$ of the circle with center $Q$ are equal. If $\angle PAB = 40^{\circ}$,then find $\angle CQD$. (in $^{\circ}$)

In the figure,$AB$ and $CD$ are two chords of a circle intersecting each other at point $E$. Prove that $\angle AEC = \frac{1}{2}$ (angle subtended by arc $CXA$ at the centre $+$ angle subtended by arc $DYB$ at the centre).

$AB$ and $CD$ are two parallel chords of a circle with centre $P$. Also,the centre $P$ lies between the chords $AB$ and $CD$. If $AB = 20\,cm$,$CD = 48\,cm$,and the radius of the circle is $26\,cm$,find the distance between $AB$ and $CD$.

In a cyclic quadrilateral $ABCD$,$\angle A : \angle C = 4 : 5$ and $\angle B : \angle D = 5 : 7$. Find all the angles of $ABCD$.