How should we prove that two circles can intersect at two points( at least)?

Question

Assume that there are two distinct circles with centres C and D respectively. I feel that these two circles can intersect at two points but I don't know how to prove that they can intersect at two points! However I tried to prove it by construction like this- "I firstly construct a circle and then I again construct other circle with a compass such that they both intersect each other at two points." Is my way of proving correct? If not,then please provide an appropriate proof for this? THANKS!

Answer 1

I think I would draw the line through $C$ and $D$. Then draw the circle centered at $D$, or at least an arc of it that cuts the segment $CD$ at $P$. Set the compass for the radius of the circle centered at $C$, but draw the circle of that radius centered at $P$. If $C$ is inside that circle, then there are two intersection points.

Answer 2

The key point to a proof is that if you have three non-collinear points, they determine a unique circle. (So two distinct circles can intersect in at most two points.) You can prove this by construction: The center of the circle will be the intersection of the perpendicular bisectors of the segments joining pairs of the points.