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P,B,Q,R lie on a circle

Source: Russia 2000 10.3

December 25, 2009
geometryparallelogramtrigonometrygeometric transformationreflectioncircumcircleangle bisector

Problem Statement

In an acute scalene triangle ABCABC the bisector of the acute angle between the altitudes AA1AA_1 and CC1CC_1 meets the sides ABAB and BCBC at PP and QQ respectively. The bisector of the angle BB intersects the segment joining the orthocenter of ABCABC and the midpoint of ACAC at point RR. Prove that PP, BB, QQ, RR lie on a circle.