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Math Help - Radius and Tangent

  1. #1
    Senior Member I-Think's Avatar
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    Radius and Tangent

    Good evening

    Does anyone here know a geometric proof for the statement

    The radius of a circle is perpendicular to a tangent to the circle at the point of tangency.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by I-Think View Post
    The radius of a circle is perpendicular to a tangent to the circle at the point of tangency.
    There can be several different proofs for this.
    Each one will depend upon on the set of axioms and theorems in use.
    That said, I can suggest some common facts that may help you.

    With the exception of the point of tangency, all other points of a tangent are in the exterior of the circle.
    The center of the circle is not on a tangent, so there is a unique perpendicular from the center to the tangent.
    You want to show that the line determined by the center and the point of tangency is that unique perpendicular.
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  3. #3
    Senior Member I-Think's Avatar
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    Indirect vs geometric proof

    I have discovered one indirect proof, following from your line of reasoning. A line perpendicular to the tangent from the origin O exists.
    As all other points on the tangent do not touch the circle, it follows that of all the lines from the origin to the tangent, the radius must be the shortest.
    As we know that the distance of a line is shortest from a point when its perpendicular distance is measured, it follows that the radius is perpendicular to the tangent at the point of tangency.

    However, I was hoping that I could be directed to a proof that is more geometric in flavor. One where the statement is proven using the angular properties of the circle.

    If such a proof exists, I would be glad if someone could lend me it.
    Thank you in advance.
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