Two Conditions for a Tetrahedron to Be Orthocentric
An altitude of a tetrahedron is a line from a vertex perpendicular to the face opposite that vertex. A tetrahedron is orthocentric if the four altitudes meet at the same point, which is called the orthocenter or the Monge point.[more]
Let the opposite side lengths of a tetrahedron be and , and and and . Then is orthocentric if and only if .
A bimedian of a tetrahedron is a line segment that joins the midpoints of a pair of opposite edges. A tetrahedron is orthocentric if and only if the three bimedians have equal length.[less]
The proof can be found in [1, p. 123].
 V. V. Prasolov and I. F. Sharygin, Problems in Stereometry (in Russian), Moscow: Nauka, 1989.