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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. –  Dennis Williamson Sep 17 '10 at 19:54
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up vote 5 down vote accepted

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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