Generalizing the concepts of perpendicular bisectors, angle bisectors, medians and altitudes of a triangle to 3D

A triangle is the most elementary, finite figure in the plane bounded by straight lines. Analogously, a tetrahedron is the most elementary solid in space bounded by planes.

Consider the corresponding analogues of the concepts of
1) perpendicular bisectors,
2) angle bisectors,
3) medians,
4) altitudes,
of a triangle for a tetrahedron, and investigate which of these are also concurrent for a tetrahedron.

Challenge: If true, prove it; if not, provide a counter-example.

First spend some time thinking about these, then click on each of the links above for a dynamic, interactive Cabri 3D applet to investigate the corresponding analogue in 3D.

Created by Michael de Villiers, 27 April 2012.