ABC is an acute-angled triangle. M is the midpoint of BC and P is the point on AM such that MB \equal{} MP. H is the foot of the perpendicular from P to BC. The lines through H perpendicular to PB, PC meet AB,AC respectively at Q,R. Show that BC is tangent to the circle through Q,H,R at H.
Original Formulation:
For an acute triangle ABC,M is the midpoint of the segment BC,P is a point on the segment AM such that PM \equal{} BM, H is the foot of the perpendicular line from P to BC,Q is the point of intersection of segment AB and the line passing through H that is perpendicular to PB, and finally, R is the point of intersection of the segment AC and the line passing through H that is perpendicular to PC. Show that the circumcircle of QHR is tangent to the side BC at point H. geometrycircumcircletrigonometrycyclic quadrilateralIMO Shortlist