Question

Prove that two circles cannot intersect at more than two points

Answer 1

Let there be two circles which intersect at three points say at $A,B$ and $C$.

Clearly $A,B$ and $C$ are not collinear.

We know that through three non-collinear points $A,B$ and $C$ one and only one circle can pass.

Therefore there cannot be two circles passing through $A,B$ and $C$. In other words the two circles cannot intersect at more than two points

Observe the following figure-