Prove that two different circles cannot intersect each other at more than two points.

Suppose two circles intersect in three points A,B,C, 

Then A,B,C are non-collinear. So, a unique circle passes through these three points. This is contradiction to the face that two given circles are passing through A,B,C. Hence, two circles cannot intersect each other at more than two points. 

Consider two different circles intersecting at three point A, B and C

We know that these points are non collinear and a unique circle can be drawn using these points

This shows that our assumption is wrong

Therefore, it is proved that two different circles cannot intersect each other at more than two points.

Suppose two circles intersect in three points A, B and C. 

Then,

 A, B, C are non-collinear. 

So, 

A unique circle passes through these three points. 

This is contradiction to the face that two given circles are passing through A, B, C. 

Hence, 

Two circles cannot intersect each other at more than two points.

If possible, let two different circles intersect at three distinct points A,B,C. Then, these points are clearly noncollinear . So, a unique circle can be drawn to pass through these points.This is a contradiction.