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Math Help - Proving . . .

  1. #1
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    Proving . . .

    Circle k(S; r)is touching point A of line AB. Circle l(T; s)is touching point B of line AB and intersects circle k in the edge points C, D of its diameter. Prove that the intersection M of lines CD and AB is the centre of line AB.
    Last edited by x-mather; January 8th 2010 at 01:09 PM.
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  2. #2
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    Trying to help but...

    Hi!
    Reading the problem over and over and canīt quite grash it. C;D are antipods of large circle? And inside that circle are two other circles?
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  3. #3
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    See this (I know it is not accurate but it helps)

    Last edited by x-mather; January 10th 2010 at 06:41 AM.
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  4. #4
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    If you know it, written form will be sufficient for me.
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  5. #5
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    Here is a sketch of the geometry.

    The large blue circle has the same radius as circle k.
    Using the radius of the smaller circle,
    next draw right-angled triangles to show |TF|=|TG|.

    The show |BM|=|BA|

    I had to shrink my sketch and it has become a little skewed.
    Those lines are meant to be perpendicular.

    This only works if CD is a centreline.
    Attached Thumbnails Attached Thumbnails Proving . . .-x-mather.jpg  
    Last edited by Archie Meade; January 10th 2010 at 07:36 AM.
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  6. #6
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    Well, does this make sence

    Is this somewhat understandable?
    Attached Thumbnails Attached Thumbnails Proving . . .-cirklar.bmp  
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  7. #7
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    Simplest is the following...

    Triangle TBE is isosceles,
    from this you can show |BM|=|MA|

    Again, CD needs to be a centreline of the smaller circle.
    Attached Thumbnails Attached Thumbnails Proving . . .-x-mather2.jpg  
    Last edited by Archie Meade; January 10th 2010 at 07:37 AM.
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  8. #8
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    A general proof, when CD is not a diameter of the smaller circle
    can be derived from the attached sketch.
    Attached Thumbnails Attached Thumbnails Proving . . .-x-mather3.jpg  
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  9. #9
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    Here is a geometric proof.
    Attached Thumbnails Attached Thumbnails Proving . . .-x-mather4.jpg  
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