Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

Two circles of radii $5\, cm$ and $3\, cm$ intersect at two points and the distance between their centres is $4\, cm.$ Find the length of the common chord. (in $, cm$)

Let the vertex of an angle $\angle ABC$ be located outside a circle and let the sides of the angle intersect equal chords $AD$ and $CE$ with the circle. Prove that $\angle ABC$ is equal to half the difference of the angles subtended by the chords $DE$ and $AC$ at the centre.

In the figure,$\angle PQR = 100^{\circ}$,where $P, Q$ and $R$ are points on a circle with centre $O$. Find $\angle OPR$. (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.

In any triangle $ABC$,if the angle bisector of $\angle A$ and the perpendicular bisector of $BC$ intersect,prove that they intersect on the circumcircle of the triangle $ABC$.