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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?

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    +1 - I wonder if there are any interesting mechanical constructions that make use of its properties.2010-09-17

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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).

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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

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It does generalizes to a sphere. When the radius of the sphere is infinite, then we get the planar case. enter image description here