I am obsessed with the nine point circle. I was thinking, is there a generalisation to aribtrary tetrahedra and spheres? What about higher dimensions? For each face of the tetrahedron, there is a nine point circle. Do these circles all lie on a sphere?
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:
There is also this paper that deals with sphere analogues: