The 24-Point Sphere

In general, the altitudes of a tetrahedron are [i]not[/i] concurrent. However, in an [b]orthocentric tetrahedron[/b], they are. There is also a 3-dimensional version of the 9-point circle called the [b]24-point sphere[/b]. This sphere contains the 9-point circle of each face, which means for each face, the midpoints of the edges, the feet of the (face) altitudes, and the (face) Euler points are on the sphere. But isn't this 36 points? Yes, but notice the midpoints of the edges and the feet of the altitudes are shared by two triangles, so there are 12 distinct points there. The Euler points of each face account for the remaining 12 points.

 

Steve Phelps

 
Resource Type
Activity
Tags
3d  geometry 
Target Group (Age)
14 – 18
Language
English (United States)
 
 
GeoGebra version
5.0
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6351
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