Two circles intersect at two points $A$ and $B$. $AD$ and $AC$ are diameters to the two circles (see Fig.). Prove that $B$ lies on the line segment $DC$.

Fill in the blanks:
$(i)$ The centre of a circle lies in . . . . . . of the circle. (exterior/ interior)
$(ii)$ $A$ point,whose distance from the centre of a circle is greater than its radius,lies in . . . . . . of the circle. (exterior/ interior)
$(iii)$ The longest chord of a circle is a . . . . . . of the circle.
$(iv)$ An arc is a . . . . . . when its ends are the ends of a diameter.
$(v)$ Segment of a circle is the region between an arc and . . . . . . of the circle.
$(vi)$ $A$ circle divides the plane,on which it lies,in . . . . . . parts.

$A$ chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

If a line intersects two concentric circles (circles with the same centre) with centre $O$ at $A, B, C$ and $D$,prove that $AB = CD$.

Prove that if chords of congruent circles subtend equal angles at their centres,then the chords are equal.

In the figure,$A, B$ and $C$ are three points on a circle with centre $O$ such that $\angle BOC = 30^{\circ}$ and $\angle AOB = 60^{\circ}$. If $D$ is a point on the circle other than the arc $ABC$,find $\angle ADC$. (in $^{\circ}$)