Points Symmetric to the Orthocenter with Respect to the Sides of a Triangle
Let ABC be a triangle. The points A', B', and C' symmetric to the orthocenter H with respect to the sides of ABC lie on the circumcircle. (A'', B'', and C'' are the feet of the perpendiculars from H to BC, CA, and AB, respectively.)
See problem 5.9 in V. Prasolov, Problems in Plane and Solid Geometry, Vol. 1, Plane Geometry [PDF], (D. Leites, ed. and trans.).