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Math Help - proportion similarity

  1. #1
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    proportion similarity

    I have a question, haven't been able to solve them..

    PQ and PR are tangents to a circle with centre O,and A is any point on QR..PB is the perpendicular from P to OA produced and meets it at B..prove that the points O,Q,B,P,R,are concyclic..if QR or RQ produced meets PB produced at C,prove that qr is divided harmonically at A and C..
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  2. #2
    MHF Contributor red_dog's Avatar
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    For the first part:
    In quadrilateral PROQ, m(\widehat{PRO})=m(\widehat{PQO})=90^{\circ}\Right  arrow PROQ is inscribable.
    In quadrilateral POQB, m(\widehat{PQO})=m(\widehat{PBO})=90^{\circ}\Right  arrow POQB is inscribable.
    The two quadrilaterals have in common the points P,O,Q, so all the five points are cyclic.
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