# Proof with circles

• May 3rd 2008, 07:41 AM
natester
Proof with circles
prove that if P is a point of the circle C, then there is just one line that is tangent to C at P.
• May 3rd 2008, 07:45 AM
Moo
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) ?
• May 3rd 2008, 07:49 AM
Isomorphism
Quote:

Originally Posted by natester
prove that if P is a point of the circle C, then there is just one line that is tangent to C at P.

Depends on how a tangent is defined to you. The equation of the tangent is

$y - y_{P} = \left(\frac{dy}{dx}\right)_{P} (x - x_{P})$

It is clear that $(x_P,y_P)$ and $\left(\frac{dy}{dx}\right)_{P}$ is fixed for a point P. Hence the tangent is unique.
• May 3rd 2008, 07:58 AM
Moo
Quote:

Originally Posted by Isomorphism
Depends on how a tangent is defined to you. The equation of the tangent is

$y - y_{P} = \left(\frac{dy}{dx}\right)_{P} (x - x_{P})$

It is clear that $(x_P,y_P)$ and $\left(\frac{dy}{dx}\right)_{P}$ is fixed for a point P. Hence the tangent is unique.

Yop,

It's in "Geometry" :/
Oh well... I don't know if he put it in the right section, now, I'm lost >_<
• May 3rd 2008, 08:00 AM
Isomorphism
Quote:

Originally Posted by Moo
Yop,
It's in "Geometry" :/
Oh well... I don't know if he put it in the right section, now, I'm lost >_<

But it's advanced geometry, so I thought derivatives will work. Anyway your argument is wonderful (Rock)
• May 3rd 2008, 09:22 AM
natester
Both of the responses helped. Thank you