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 $DC$.

In the figure,$\angle ABC = 69^{\circ}$ and $\angle ACB = 31^{\circ}$. Find $\angle BDC$. (in $^{\circ}$)

In the figure,$\angle PQR = 100^{\circ}$,where $P, Q$ and $R$ are points on a circle with centre $O$. Find $\angle OPR$. (in $^{\circ}$)

Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

$AC$ and $BD$ are chords of a circle which bisect each other. Prove that $AC$ and $BD$ are diameters.

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.