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tangent line proof in KJMO 2006

Source: KJMO 2006 p7

May 1, 2019
geometrycircletangentperpendicular

Problem Statement

A line through point PP outside of circle OO meets the said circle at B,CB,C (PB<PCPB < PC). Let POPO meet circle OO at Q,DQ,D (with PQ<PDPQ < PD). Let the line passing QQ and perpendicular to BCBC meet circle OO at AA. If BD2=ADā‹…CPBD^2 = AD\cdot CP, prove that PAPA is a tangent to OO.
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