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 $AC$ are two chords of a circle with centre $P$. If the bisector of $\angle BAC$ passes through the centre $P$,prove that $AB = AC$.

$AB$ and $CD$ are two parallel chords of a circle with centre $P$. Also,the centre $P$ lies between the chords $AB$ and $CD$. If $AB = 20\,cm$,$CD = 48\,cm$,and the radius of the circle is $26\,cm$,find the distance between $AB$ and $CD$.

In a circle with centre $O$,two chords $PQ$ and $RS$ subtend equal angles at the centre. If $PQ = 12 \text{ cm}$,then $RS = \dots \text{ cm}$.

In a cyclic quadrilateral $ABCD$,$\angle A = 4 \angle C$ and $\angle B = 3 \angle D$. Find all the angles of $ABCD$.

Prove that two circles cannot intersect at more than two points.