If i want to prove that a point $O$ is a center of a circle. is it sufficient to say that if $A,B,C$ are points On the circle and $AO=BO=CO$ so point $O$ is the center because of: Through any three points, Not all on the same line, there lies a unique circle.
Or the last sentence isn't sufficient to say that $O$ is the center and i need something else to connect between them and prove it?
If I understood correctly the question, the answer is yes: a circle is uniquely determined by three non-collinear points $\,A,B,C\,$ and a fourth, different one $\;O\;$ which is at the same distance from each of the first three.
The proof is easy: form the triangle $\;\Delta ABC\;$, then the circle $\,O\;$ is this triangle's circumcircle...If you're not sure about this you can try drawing the perpendicular bisector of any two pairs of these three points, say of $\;AB\,,\,BC\;$ . As any point on the perp. bisector of $\;AB\;$ is at the same distance from $\,A\,$ and from $\,B\,$, and every point on the p.b. of $\,BC\;$ is at the same distance from $\,B\,$ and from $\,C\,$ , the intersection of these two p.b.'s (why do they have to intersect?!) is precisely $\,O\,$ ...