Two Conditions for a Tetrahedron to Be Orthocentric

a

1.2

b

1

c

1

d

1.2

2D/3D

net	polyhedron

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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

T

be

a

and

e

,

b

and

f

and

c

and

d

. Then

T

is orthocentric if and only if

2

a

+

2

e

=

2

b

+

2

f

=

2

c

+

2

d

.

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.