Abstract:

Previous articles have discussed about the properties of orthocentric tetrahedrons: nine-point circles on each face cospherical and the 3D Euler line. This paper aims at finding the sufficient and necessary conditions for the nine-point circles to be cospherical in the triangular polyhedrons. First, we discussed the conditions for the nine-point circles to be cospherical in a tetrahedron, in a hexahedron and in an octahedron. Next, we found that the 3Dorthocenter

* H*_{c} , the center of the 24-point sphere (48-point sphere)

* N*_{c} and the 3D circumcenter O

_{c} of a tetrahedron (an octahedron), if they exist, must be collinear and the ratio of the distance between them is

* H*_{cNc : NcOc} = 1 : 1. After studying the properties of triangular polyhedrons, we have found that the existence of the 3D orthocenter and the 3D circumcenter is the necessary condition for the nine-point circles to be cospherical.

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