Centroid and Incenter in Isosceles Tetrahedra

Statement

Centroid and Incenter in Isosceles Tetrahedra

Condition $G=I\,$ is necessary

We know that in an isosceles tetrahedron $G=O\,$ and $O=I.\;$ By transitivity, $G=I.$

Condition $G=I\,$ is sufficient

The barycentric coordinates have been discussed at this site only in the plane, relative to a triangle. But the same ideas work in 3D. For example, the centroid has (homogeneous) barycentric coordinates $1:1:1:1\,$ whereas the incenter is identified by $a:b:c:d,\,$ where $a,b,c,d\,$ are the areas of the faces of the tetrahedron. For the two centers to coincide, their coordinates need to be proportional which, in this case, requires the tetrahedron to be equiareal, i.e., to have all faces of the same area. But it's known that equiareal tetrahedra are also isosceles.

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