Question

prove that two different circles cannot each other at more than two points

Answer 1

Here's a great way without any coordinates.

Assume there existed three points at which the circles intersected, say A,B,C.  Invert at A with arbitrary radius.  The circles map to lines which intersect at both B' and C' (as well as A', the point at infinity).  Since any two different lines meet at at most one point, the lines are the same, so the circles are the same.  Thus, we don't have two circles, contradiction.