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Prove that two different circles cannot intersect each other at more than two points.
We have to prove that two different circles cannot intersect each other at more than two points.
Let two circles intersect each other at three points $A, B$ and $C$
Two circles with centres $O$ and $O’$ intersect at $A, B$ and $C$
$A, B$ and $C$ are non-collinear points.
Circle with centre $O$ passes through three points $A, B$ and $C$ and circle with centre $O’$ also passes through three points $A, B$ and $C$
One and only one circle can be drawn through three points.
Therefore, our supposition is wrong
Two circles cannot intersect each other not more than two points.
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