There is one and only one circle passing through all the vertices $A, B$ and $C$ of $\Delta ABC$. This circle is called the ............. of $\Delta ABC$.

$AB$ and $CD$ are two parallel chords of a circle with centre $P$. Also,the centre $P$ lies between $AB$ and $CD$. If $AB = 48 \, cm$,$CD = 40 \, cm$,and the radius of the circle is $25 \, cm$,find the distance between $AB$ and $CD$. (in $, cm$)

If a line segment joining the mid-points of two chords of a circle passes through the centre of the circle,prove that the two chords are parallel.

In a cyclic quadrilateral $ABCD$,$\angle A = 2x - 10^{\circ}$ and $\angle C = 3x - 35^{\circ}$,then $\angle A =$ .......... (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 a cyclic quadrilateral $ABCD$,$\angle A : \angle C = 4 : 5$ and $\angle B : \angle D = 5 : 7$. Find all the angles of $ABCD$.