Draw different pairs of circle how …

From the figure, we observe that when different pairs of circles are drawn, each pair have two points (say A and B) in common.
Maximum number of common points are two in number.

Suppose two circles C (O) and C (O’) intersect each other in three points, say A, B and C.
Then A, B and C are non-collinear points.
We know that:
There is one and only one circle passing through three non-collinear points.
Therefore, a unique circle passes through A, B and C.
{tex} \Rightarrow {/tex} O’ coincides with O and s = r where s and r are the radii of two circles C(O) and C(O')
A contradiction to the fact that C (O’,) {tex} \ne {/tex} C (O,)
{tex}\therefore {/tex} Our supposition is wrong.
Hence two different circles cannot intersect each other at more than two points.