geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, an orthocentric tetrahedron is a

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...

where all three pairs of opposite edges are

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...

. It is also known as an orthogonal tetrahedron since orthogonal means perpendicular. It was first studied by

Simon Lhuilier Simon Antoine Jean L'Huilier (or L'Huillier) (24 April 1750 in Geneva – 28 March 1840 in Geneva) was a Swiss mathematician of French Huguenot descent. He is known for his work in mathematical analysis and topology, and in particular the ...

in 1782, and got the name orthocentric tetrahedron by G. de Longchamps in 1890.. In an orthocentric tetrahedron the four altitudes are concurrent. This common point is called the orthocenter, and it has the property that it is the symmetric point of the center of the

circumscribed sphere In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle' ...

with respect to the

centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...

. Hence the orthocenter coincides with the Monge point of the tetrahedron.

Characterizations

All tetrahedra can be inscribed in a

parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclid ...

. A tetrahedron is orthocentric

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...

its circumscribed parallelepiped is a

rhombohedron In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be us ...

. Indeed, in any tetrahedron, a pair of opposite edges is perpendicular if and only if the corresponding faces of the circumscribed parallelepiped are rhombi. If four faces of a parallelepiped are rhombi, then all edges have equal lengths and all six faces are rhombi; it follows that if two pairs of opposite edges in a tetrahedron are perpendicular, then so is the third pair, and the tetrahedron is orthocentric. A tetrahedron is orthocentric if and only if the sum of the squares of opposite edges is the same for the three pairs of opposite edges:Reiman, István, "International Mathematical Olympiad: 1976-1990", Anthem Press, 2005, pp. 175-176. :

\overline^2 + \overline^2 = \overline^2 + \overline^2 = \overline^2 + \overline^2.

In fact, it is enough for only two pairs of opposite edges to satisfy this condition for the tetrahedron to be orthocentric. Another necessary and sufficient condition for a tetrahedron to be orthocentric is that its three bimedians have equal length.

Hazewinkel, Michiel Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician, and Emeritus Professor of Mathematics at the Centre for Mathematics and Computer Science and the University of Amsterdam, particularly known for his 1978 book ''Formal groups and ...

, "Encyclopaedia of mathematics: Supplement, Volym 3", Kluwer Academic Publishers, 1997, p. 468.

Volume

The characterization regarding the edges implies that if only four of the six edges of an orthocentric tetrahedron are known, the remaining two can be calculated as long as they are not opposite to each other. Therefore the

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...

of an orthocentric tetrahedron can be expressed in terms of four edges ''a'', ''b'', ''c'', ''d''. The formula isAndreescu, Titu and Gelca, Razvan, "Mathematical Olympiad Challenges", Birkhäuser, second edition, 2009, pp. 30-31, 159. :

V=\frac\sqrt

where ''c'' and ''d'' are opposite edges, and

s=\tfrac(a+b+c)

References

{{reflist Polyhedra

Characterizations

Volume

See also

References