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Triangle, orthocenter, parallels - prove that EX || AP

Source: IMO Shortlist 1996, G1

December 10, 2005
geometrycircumcirclereflectionvectorparallelogramIMO Shortlistmoving points

Problem Statement

Let ABC ABC be a triangle, and H H its orthocenter. Let P P be a point on the circumcircle of triangle ABC ABC (distinct from the vertices A A, B B, C C), and let E E be the foot of the altitude of triangle ABC ABC from the vertex B B. Let the parallel to the line BP BP through the point A A meet the parallel to the line AP AP through the point B B at a point Q Q. Let the parallel to the line CP CP through the point A A meet the parallel to the line AP AP through the point C C at a point R R. The lines HR HR and AQ AQ intersect at some point X X. Prove that the lines EX EX and AP AP are parallel.
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