The Center and Radius of the Nine-Point Circle
Let ABC be a triangle with orthocenter H, circumcenter O, and circumradius . Let the nine-point circle have center N and radius .[more]
Then N lies on OH (the Euler line) and bisects OH.
See theorem 209 in N. Altshiller-Court, College Geometry, Mineola, NY: Dover, 2007 p. 104.