There can be several different proofs for this.

Each one will depend upon on the set of axioms and theorems in use.

That said, I can suggest some common facts that may help you.

With the exception of the point of tangency, all other points of a tangent are in the exterior of the circle.

The center of the circle is not on a tangent, so there is a unique perpendicular from the center to the tangent.

You want to show that the line determined by the center and the point of tangency is that unique perpendicular.