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
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2
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2
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2
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2
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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.