The length of the complete circle is called its ............

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.

State whether each of the following statements is true or false:
$(1)$ For any two points on a circle,the line segment joining them is called a chord of the circle.
$(2)$ The length of a diameter of a circle is half the length of its radius.

$A, B$ and $C$ are three points on a circle. Prove that the perpendicular bisectors of $AB, BC$ and $CA$ are concurrent.

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

In the figure,$\angle OAB = 30^{\circ}$ and $\angle OCB = 57^{\circ}$. Find $\angle BOC$ and $\angle AOC$.