In the figure,$ABCD$ is a cyclic quadrilateral in which $AC$ and $BD$ are its diagonals. If $\angle DBC = 55^{\circ}$ and $\angle BAC = 45^{\circ}$,find $\angle BCD$. (in $^{\circ}$)

If two circles intersect at two points,prove that their centres lie on the perpendicular bisector of the common chord.

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.

If two equal chords of a circle intersect within the circle,prove that the segments of one chord are equal to corresponding segments of the other chord.

Two congruent circles intersect each other at points $A$ and $B$. Through $A$,any line segment $PAQ$ is drawn such that $P$ and $Q$ lie on the two circles. Prove that $BP = BQ$.

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 $^{\circ}$)