10
$\begingroup$

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?

$\endgroup$
1
  • 1
    $\begingroup$ +1 - I wonder if there are any interesting mechanical constructions that make use of its properties. $\endgroup$
    – Dennis Williamson
    Sep 17, 2010 at 19:54

2 Answers 2

Reset to default
6
$\begingroup$

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).

$\endgroup$
4
$\begingroup$

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

$\endgroup$
1
  • $\begingroup$ The link to springerlink.com is broken, and I'm also unable to find any copy saved on the Wayback Machine... $\endgroup$
    – The Amplitwist
    Apr 16, 2022 at 20:24

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .