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Math Help - Circles - Please Help

  1. #1
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    Exclamation Circles - Please Help

    Two circles touch internally at a point A and the smaller of the two circles passes through O, the centre of the larger circle. AB is any chord of the larger circle, cutting the smaller circle S. The tangents to the larger circle at A and B meet at a point T.



    Prove:

    i) AB is bisected at S.

    ii) O, S and T are collinear.

    Could someone please help me get started?
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  2. #2
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    Hello, xwrathbringerx!

    Two circles touch internally at a point A
    and the smaller of the two circles passes through O, the centre of the larger circle.
    AB is any chord of the larger circle, cutting the smaller circle S.
    The tangents to the larger circle at A and B meet at a point T.



    Prove:

    a) AB is bisected at S.

    b) O, S\text{ and }T are collinear.

    Draw OT.
    Draw radii OA = OB = r

    TA = TB
    Tangents to a circle from an extrenal point are equal.

    Points O and T are equidistant from points A and B.
    . . Hence, OT is the perpendicular bisector of AB.


    OA is a diameter of the small circle.
    \angle OSA is inscribed in a semicircle: . \angle OSA = 90^o

    Hence, S lies on OT.


    And the two proofs follow . . .

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  3. #3
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    Can I simply say OA is a diameter of the small circle?
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