Problem

Source: Iran TST 2006

Tags: geometry, parallelogram, circumcircle, geometric transformation, reflection, conics, angle bisector



Let l,m be two parallel lines in the plane. Let P be a fixed point between them. Let E,F be variable points on l,m such that the angle EPF is fixed to a number like α where 0<α<π2. (By angle EPF we mean the directed angle) Show that there is another point (not P) such that it sees the segment EF with a fixed angle too.