Four Concyclic Points
Let ABC be a triangle and H the foot of the altitude from B. Let M be the midpoint of AC. Let P and Q be the feet of the perpendiculars from A and C to the angle bisector of the angle ABC. Then H, M, P and Q lie on a circle.
D. Grinberg, "From Baltic Way to Feuerbach: A Geometrical Excursion," Mathematical Refections (2), 2000 p. 2.