The 24-Point Sphere
- Steve Phelps
In general, the altitudes of a tetrahedron are not concurrent. However, in an orthocentric tetrahedron, they are. There is also a 3-dimensional version of the 9-point circle called the 24-point sphere. 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.