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 $^{\circ}$)

Bisectors of angles $A, B$ and $C$ of a triangle $ABC$ intersect its circumcircle at $D, E$ and $F$ respectively. Prove that the angles of the triangle $DEF$ are $90^{\circ} - \frac{1}{2}A, 90^{\circ} - \frac{1}{2}B$ and $90^{\circ} - \frac{1}{2}C$.

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 $^{\circ}$)

$A$ circular park of radius $20 \, m$ is situated in a colony. Three boys Ankur,Syed and David are sitting at equal distances on its boundary,each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.

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.

$A$ chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.