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Math Help - problem on circle theorems

  1. #1
    Member smmmc's Avatar
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    problem on circle theorems

    PQ is a variable chord of the smaller of two fixed concentric circles.
    PQ produced meets the circumference of the larger circle at R. Prove that the product
    RP.RQ is constant for all positions and lengths of PQ.

    How would you prove this?

    thanks
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  2. #2
    MHF Contributor
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    Hi

    First you can take some simple examples to have an idea of the value of the constant
    The easiest one is to take PQ as a diameter of the smaller circle

    Then considering the generic case



    Let O be the center of the two circles and I the center of [PQ]

    \overrightarrow{RP}\cdot\overrightarrow{RQ}=RP\:RQ

    \overrightarrow{RP}\cdot\overrightarrow{RQ}=\left(  \overrightarrow{RI}+\overrightarrow{IP}\right)\cdo  t\left(\overrightarrow{RI}+\overrightarrow{IQ}\rig  ht)

    Expanding and simplifying

    \overrightarrow{RP}\cdot\overrightarrow{RQ}=RI^2-IP^2

    \overrightarrow{RP}\cdot\overrightarrow{RQ}=(RO^2-OI^2)-IP^2=RO^2-(OI^2+IP^2)=RO^2-OP^2

    (Properties of right angle triangles)

    Therefore RP.RQ is equal to the difference of the squares of the 2 radii
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