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Math Help - Circles

  1. #1
    Senior Member
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    Exclamation Circles

    Two circles touch internally at A. The tangent at P on the smaller circle cuts the larger circle at Q and R. Prove that AP bisects angle RAQ.

    Pleas help?
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  2. #2
    Super Member malaygoel's Avatar
    Joined
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    This one can also be done with the help of Alternate segment theorem

    Let AR intersect smaller circle at N
    Let AQ intersect smaller circle at M
    Join MN

    Draw a tangent a A
    Let Y be the point on tangent such that angle RAY is acute
    Let Z be the point on tangent such that angle RAY is obtuse

    The following relation follows from Alternate Segment theorem
    RAY=RQA
    PAY=PMA
    QAZ=QRA
    PAZ=PNA

    PAQ=PAZ-QAZ
    =PNA-QRA
    =RPN
    =PAR(Alternate ST,smaller circle, tangent at P, chord PN)

    Hence, proved.
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