Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

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.

$ABCD$ is a parallelogram. The circle passing through $A, B$ and $C$ intersects $CD$ (produced if necessary) at $E$. Prove that $AE = AD$.

In any triangle $ABC$,if the angle bisector of $\angle A$ and the perpendicular bisector of $BC$ intersect,prove that they intersect on the circumcircle of the triangle $ABC$.

Bisectors of angles $A, B$ and $C$ of a triangle $ABC$ intersect its circumcircle at $D, E$ and $F$ respectively. Prove that the angles of the triangle $DEF$ are $90^{\circ} - \frac{1}{2}A, 90^{\circ} - \frac{1}{2}B$ and $90^{\circ} - \frac{1}{2}C$.

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