There is one and only one circle passing through all the vertices $A, B$ and $C$ of $\Delta ABC$. This circle is called the ............. of $\Delta ABC$.

State whether the following statements are True or False. Give reasons for your answers.
$(i)$ The line segment joining the centre to any point on the circle is a radius of the circle.
$(ii)$ $A$ circle has only a finite number of equal chords.
$(iii)$ If a circle is divided into three equal arcs,each is a major arc.
$(iv)$ $A$ chord of a circle,which is twice as long as its radius,is a diameter of the circle.
$(v)$ $A$ sector is the region between the chord and its corresponding arc.
$(vi)$ $A$ circle is a plane figure.

The longest chord of a circle is a .......... of the circle.

Write True or False and justify your answer in each of the following:
Two chords $AB$ and $AC$ of a circle with centre $O$ are on the opposite sides of $OA$. Then $\angle OAB = \angle OAC$.

Two equal chords $AB$ and $CD$ of a circle when produced intersect at a point $P$. Prove that $PB = PD$.

$AB$ and $AC$ are two chords of a circle of radius $r$ such that $AB = 2 AC$. If $p$ and $q$ are the distances of $AB$ and $AC$ from the centre,prove that $4 q^{2} = p^{2} + 3 r^{2}$.