Orthocentric tetrahedron

It was first studied by Simon Lhuilier in 1782, and got the name orthocentric tetrahedron by G. de Longchamps in 1890.

It has the property that it is the symmetric point of the center of the circumscribed sphere with respect to the centroid.

Indeed, in any tetrahedron, a pair of opposite edges is perpendicular if and only if the corresponding faces of the circumscribed parallelepiped are rhombi.

If four faces of a parallelepiped are rhombi, then all edges have equal lengths and all six faces are rhombi; it follows that if two pairs of opposite edges in a tetrahedron are perpendicular, then so is the third pair, and the tetrahedron is orthocentric.

Another necessary and sufficient condition for a tetrahedron to be orthocentric is that its three bimedians have equal length.