Does the nine point circle generalise to some theorem about n-spheres and n-simplices?

An orthocentric tetrahedron is one for which the altitudes from the vertices to the opposite faces are concurrent (this is not true for all tetrahedrons). For an orthocentric tetrahedron, there exists a sphere (the 24-point sphere) that intersects each face of the tetrahedron in its 9-point circle.

See also the summary of a talk by Steve West ("Discovering Theorems Using Cabri 3-D") in the November 2008 issue of Points & Angles (PDF).

Take a look at:

http://www.springerlink.com/content/mj55u41710w625j8/

and

http://en.wikipedia.org/wiki/Tetrahedron

There is also this paper that deals with sphere analogues:

http://www.jstor.org/pss/3617834