IMO Shortlist 1988 problem 15
Let be an acute-angled triangle. The lines , and are constructed through the vertices , and respectively according the following prescription: Let be the foot of the altitude drawn from the vertex to the side ; let be the circle with diameter ; let meet the sides and at and respectively, where and are distinct from ; then let be the line through perpendicular to . The lines and are constructed similarly. Prove that the lines , and are concurrent.