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 $^{\circ}$)

In the figure,$AB$ is a diameter of the circle,and $CD$ is a chord equal to the radius of the circle. $AC$ and $BD$ when extended intersect at a point $E$. Prove that $\angle AEB = 60^{\circ}$.

If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection,prove that the chords are equal.

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 $ABC$.

Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

Let the vertex of an angle $\angle ABC$ be located outside a circle and let the sides of the angle intersect equal chords $AD$ and $CE$ with the circle. Prove that $\angle ABC$ is equal to half the difference of the angles subtended by the chords $DE$ and $AC$ at the centre.