The Refelctions of a Line Through the Orthocenter
Take any line through the orthocenter H of the triangle (drag point E). Reflect this line in the sides of the triangle.
The reflections are concurrent at a point (F) on the circumcircle.
Take this point F and reflect it in the side of the triangle. Its reflections lie on the give line through the orthocenter.
This is a theorem attributed to Collings and Longuet-Higgins.