proof that triangle have only one particular circumscribed circle

Question

[What I know]

1. I know that a triangle is cyclic polygon and every triangle has a circumscribed circle.

2. I know that every triangle has only one circumcenter point. since the definition of circumcenter is "The point where the three perpendicular bisectors of a triangle meet." I am not sure that this definition necessarily means that the circumcenter of the triangle is equal to the center of the triangle's circumscribed circle.

[What I want to know]

I want to know the rigorous proof that shows that there is only "one" circle that circumscribes a given triangle.

Answer 1

Let $\Delta ABC$ be our triangle, $\Phi$ be another circle and let $O$ be a center of the circle.

Thus, $OA=OB$, which says that $O$ is placed on the perpendicular bisector to $AB$.

By the same way we obtain that $O$ is placed on the perpendicular bisector to $AC$.

But, you said that you know that these perpendicular bisectors intersects in your center of your circle, say $T$.

But since two different lines have an unique common point, we obtain $O\equiv T$.

Can you end it now?