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circumcircle and altitudes! CMO 2015 P4

Source: canadian mathematical olympiad

April 24, 2015
geometrycircumcircleCanada2014National Olympiads

Problem Statement

Let ABCABC be an acute-angled triangle with circumcenter OO. Let II be a circle with center on the altitude from AA in ABCABC, passing through vertex AA and points PP and QQ on sides ABAB and ACAC. Assume that BP⋅CQ=AP⋅AQ.BP\cdot CQ = AP\cdot AQ. Prove that II is tangent to the circumcircle of triangle BOCBOC.
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