Given a circle and a point outside of the circle,
Connect the point, A, with the center of the circle, O
Let M be the midpoint of OA. Draw the circle centered at M going through A and O.
Let the point where the two circles meet be C. Connect AC.
AC is a line through A tangent to the circle.
Top see that AC is tangent to the circle, connect OC
Angle ACO is an inscribed angle in the circle about M, so the inscribed angle ACO is half of the central angle which is the diameter AMO. So OC is perpendicular to AC. Since the tangent is perpendicular to the radius to the point of tangency, by the uniqueness of the line through C perpendicular to OC, AC is tangent to the circle.