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points equally distant from a point on the altitude

Source: MEMO 2016 T5

August 25, 2016
geometrycircumcirclegeometry proposedLaw of Cosines

Problem Statement

Let ABCABC be an acute triangle for which AB≠ACAB \neq AC, and let OO be its circumcenter. Line AOAO meets the circumcircle of ABCABC again in DD, and the line BCBC in EE. The circumcircle of CDECDE meets the line CACA again in PP. The lines PEPE and ABAB intersect in QQ. Line passing through OO parallel to the line PEPE intersects the AA-altitude of ABCABC in FF.
Prove that FP=FQFP = FQ.
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