Question

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

Answer 1

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.
$\therefore$These points are non-collinear points.
Three non-collinear points determine one and only one circle.
$\therefore$ 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.