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Math Help - Proof with circles

  1. #1
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    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.
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    Moo
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    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) ?
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    Quote Originally Posted by natester View Post
    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.
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    Moo
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    Quote Originally Posted by Isomorphism View Post
    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 >_<
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  5. #5
    Lord of certain Rings
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    Quote Originally Posted by Moo View Post
    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
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  6. #6
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    Both of the responses helped. Thank you
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