$A$ point located in the plane of a circle and which is equidistant from every point on the circle is called the ........ of the circle.

Prove that equal chords of a circle subtend equal angles at the centre.

In the figure,$\angle ACB = 40^{\circ}$. Find $\angle OAB$.

In a circle with centre $P$,$AB$ is a chord and $PA = 4\, cm$. In a circle with centre $Q$,$XY$ is a chord and $QX = 4\, cm$. If $\angle APB = 80^{\circ}$,$\angle XQY = 50^{\circ}$ and $AB = 5\, cm$,then $XY = \dots\dots\dots\, cm$.

If $ABC$ is an equilateral triangle inscribed in a circle and $P$ is any point on the minor arc $BC$ which does not coincide with $B$ or $C$,prove that $PA$ is the angle bisector of $\angle BPC$.

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