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Math Help - Tangent to a circle!

  1. #1
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    Tangent to a circle!

    From an external point P , two tangents PA & PB are drawn to a circle with center O. Prove that OP is perpendicular bisector of AB.
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  2. #2
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    Quote Originally Posted by prantik007 View Post
    From an external point P , two tangents PA & PB are drawn to a circle with center O. Prove that OP is perpendicular bisector of AB.
    Here PA = PB
    OA is perpendicular to PA
    OB is perpendicular to PB. Hence QAPB is a cyclic quadrilateral.
    Angle OAP = ABP = 90 degrees.
    If AB cuts OP at D,angle OAD = angle OPB = angle OPA = angle OBD.
    Hence you can show that
    D is mid point of AB
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  3. #3
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    Hello prantik007
    Quote Originally Posted by prantik007 View Post
    From an external point P , two tangents PA & PB are drawn to a circle with center O. Prove that OP is perpendicular bisector of AB.
    My thanks to sa-ri-ga-ma for his contribution so far. He is quite correct up to the point where he says:
    Hence you can show that
    D is mid point of AB
    But I'm afraid that this might be a bit of a cop-out.

    I think you also need to use the fact that \angle DBP = 90^o-\angle OBD. Then use the fact that \angle BPD = \angle OBD to say that:
    \angle DBP = 90^o-\angle BPD

    \Rightarrow \angle BDP = 90^o (angle sum of \triangle BDP)
    Now we have established that AB \perp OP, it's very easy to show that \triangle's AOD and BOD are congruent, and hence that AD = DB.

    Can you fill in the details now?

    Grandad
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