prove that if P is a point of the circle C, then there is just one line that is tangent to C at P.
Do it by a proof by contradiction.
Suppose that there are 2 tangent lines D & D' to C at P.
Let O be the center of the circle.
Then D is perpendicular to OP, and so is D'.
Hence, D & D' are parallel.
Is it possible that two different & parallel lines go through a common point (P) ?