prove that if P is a point of the circle C, then there is just one line that is tangent to C at P.
Hello,
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) ?
Depends on how a tangent is defined to you. The equation of the tangent is
$\displaystyle y - y_{P} = \left(\frac{dy}{dx}\right)_{P} (x - x_{P})$
It is clear that $\displaystyle (x_P,y_P)$ and $\displaystyle \left(\frac{dy}{dx}\right)_{P}$ is fixed for a point P. Hence the tangent is unique.