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I have found a new property of the Tetrahedron. In fact, this is a 3D generalization of theorem 1 in my paper "A Note on Reflection", published by Forum Geometricorum. Consider any Tetrahedron ABCD. Take an arbitrary point P on the space. Now, reflect P around the four centroids of the four triangular faces of the Tetrahedron.Then, the line segments joining the vertices with the symmetry image of P corresponding to the opposite faces of the vertices are concurrent.

There has been discussion here :

1 https://groups.yahoo.com/neo/groups/Quadri-Figures-Group/conversations/messages/1111

[2] https://groups.yahoo.com/neo/groups/Quadri-Figures-Group/conversations/messages/1116

[3]https://groups.yahoo.com/neo/groups/Quadri-Figures-Group/conversations/messages/1117

I wonder whether this property can be generalizes to other polyhedra.enter image description here

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  • $\begingroup$ Also, is there a 3D version of Desargues Theorem? $\endgroup$ – Emmanuel García May 17 '15 at 19:23
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This is more a property of a number of points than dimensionality of the collection. It does generalize to any number of points

http://www.cut-the-knot.org/triangle/AffinePropertyOfBarycenter.shtml

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