# Two Conditions for a Tetrahedron to Be Orthocentric

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.

Let the opposite side lengths of a tetrahedron be and , and and and . Then is orthocentric if and only if .

T

a

e

b

f

c

d

T

a+e=b+f=c+d

2

2

2

2

2

2

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.