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Math Help - [SOLVED] Harmonic ratios

  1. #1
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    Exclamation [SOLVED] Harmonic ratios

    Suppose we have two orthogonal circles and any diameter AC of one circle (say) cuts the other circle at B and D.
    The circles are intersecting and the prolonged diameter of the first circle (centre O) also passes through the second centre (O'). Hint: Consider OB.OD...

    Prove that B and D divide AC harmonically...
    Help?
    Last edited by HelenaStage; September 14th 2009 at 01:04 PM.
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  2. #2
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    Hi,
    can you please restate the problem more clearly?
    I don't understand

    • why the first sentence ends with "and."
    • what is a "diagonal of a circle"
    • where's point D, maybe some drawing would be helpful to know what points are in play
    • what is actually the question? (:
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  3. #3
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    [QUOTE=Taluivren;364954]Hi,
    can you please restate the problem more clearly?
    I don't understand

    Sorry, I realize I wrote it in a hurry...should be better now.
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  4. #4
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    It is still very confusing, my guess of what's going on is on the attached picture.

    Hint tells us to use power of a point O to the circle with center O', OE^2=OB.OD

    AB = OB+OA = OB+OE = OE^2/OD +OE = (OE/OD)(OE+OD)
    CB = OC-OB = OE-OB = OE-OE^2/OD

    AB/CB = (OE+OD)/(OD-OE)

    AD = OA+OD = OE+OD
    CD = OD-OC = OD-OE

    AD/CD = (OE+OD)/(OD-OE)

    AB/CB = AD/CD.
    Attached Thumbnails Attached Thumbnails [SOLVED] Harmonic ratios-cc.jpg  
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  5. #5
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    You got it almost right, but the diameters of the circles are colinear, and you need to switch points B and C.
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  6. #6
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    Quote Originally Posted by HelenaStage View Post
    You got it almost right, but the diameters of the circles are colinear, and you need to switch points B and C.
    if you interchange points B and C on the picture, the calculation above will stay correct as it is, showing that B and D divide AC harmonically. Sorry for interchanging them on the pic.

    I've drawn the points A,O,O' not colinear on purpose, because for the calculation above we don't need their colinearity at all.
    Is everything clear or should the picture be redrawn? (:
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  7. #7
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    Quote Originally Posted by Taluivren View Post
    if you interchange points B and C on the picture, the calculation above will stay correct as it is, showing that B and D divide AC harmonically. Sorry for interchanging them on the pic.

    I've drawn the points A,O,O' not colinear on purpose, because for the calculation above we don't need their colinearity at all.
    Is everything clear or should the picture be redrawn? (:
    Oh no, I get it now!

    Thanks a lot...XD
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