Question

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

Answer 1

To prove: Two different circles cannot intersect each other at more than two points.

Construction: Let two circles intersect each other at three points $A,B$ and $C$

Proof :

Since two circles with centres $O$ and $O^{\prime }$ intersect at $A,B$ and $C$.

$\therefore A,BandC$ are non-collinear points

$\therefore$ Circle with centre $O$ passes through three points $A,B$ and $C$.

and circle with centre $O^{\prime }$ also passes through three points $A,B$ and $C$.

But one and only one circle can be drawn through three points

$\therefore$ Our supposition is wrong

$\therefore$ Two circle cannot intersect each other not more than two points.

Observe the following figure,