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 $\Delta ABC$.

In the figure,$\angle AOB = 90^{\circ}$ and $\angle ABC = 30^{\circ},$ then $\angle CAO$ is equal to: (in $^{\circ}$)

If two equal chords of a circle intersect,prove that the parts of one chord are separately equal to the parts of the other chord.

$ABCD$ is a cyclic quadrilateral such that $AB$ is a diameter of the circle circumscribing it and $\angle ADC = 140^{\circ}$,then $\angle BAC$ is equal to (in $^{\circ}$)

In a cyclic quadrilateral $PQRS$,$\angle P = \angle R + 50^{\circ}$. Find $\angle P$ and $\angle R$.

$AB$ 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 $, cm$)