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

Explanation

To prove: Two distinct circles cannot intersect each other in more than two points.

Proof: Suppose that two distinct circles intersect each other in more than two points.
So, These points are non-collinear points.

Three non-collinear points determine one and only one circle.

Since there should be only one circle. Therefore, from those three points, 2 circles cannot pass.

This contradicts the given, which shows that our assumption is wrong.

Hence, two distinct circles cannot intersect each other in more than two points.