Editing Tetrahedron (section)

===Properties analogous to those of a triangle===
The tetrahedron has many properties analogous to those of a triangle, including an insphere, circumsphere, medial tetrahedron, and exspheres. It has respective centers such as incenter, circumcenter, excenters, [[Spieker circle|Spieker center]] and points such as a centroid. However, there is generally no orthocenter in the sense of intersecting altitudes.<ref>{{Cite journal |last1=Havlicek |first1=Hans |last2=Weiß |first2=Gunter |title=Altitudes of a tetrahedron and traceless quadratic forms |journal=[[American Mathematical Monthly]] |volume=110 |issue=8 |pages=679–693 |year=2003 |url=http://www.geometrie.tuwien.ac.at/havlicek/pub/hoehen.pdf |doi=10.2307/3647851 |jstor=3647851 |arxiv=1304.0179 }}</ref>

[[Gaspard Monge]] found a center that exists in every tetrahedron, now known as the '''Monge point''': the point where the six midplanes of a tetrahedron intersect. A midplane is defined as a plane that is orthogonal to an edge joining any two vertices that also contains the centroid of an opposite edge formed by joining the other two vertices. If the tetrahedron's altitudes do intersect, then the Monge point and the orthocenter coincide to give the class of [[orthocentric tetrahedron]].

An orthogonal line dropped from the Monge point to any face meets that face at the midpoint of the line segment between that face's orthocenter and the foot of the altitude dropped from the opposite vertex.

A line segment joining a vertex of a tetrahedron with the [[centroid]] of the opposite face is called a ''median'' and a line segment joining the midpoints of two opposite edges is called a ''bimedian'' of the tetrahedron. Hence there are four medians and three bimedians in a tetrahedron. These seven line segments are all [[Concurrent lines|concurrent]] at a point called the ''centroid'' of the tetrahedron.<ref>Leung, Kam-tim; and Suen, Suk-nam; "Vectors, matrices and geometry", Hong Kong University Press, 1994, pp. 53–54</ref> In addition the four medians are divided in a 3:1 ratio by the centroid (see [[Commandino's theorem]]). The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter. These points define the ''Euler line'' of the tetrahedron that is analogous to the [[Euler line]] of a triangle.

The [[nine-point circle]] of the general triangle has an analogue in the circumsphere of a tetrahedron's medial tetrahedron. It is the '''twelve-point sphere''' and besides the centroids of the four faces of the reference tetrahedron, it passes through four substitute ''Euler points'', one third of the way from the Monge point toward each of the four vertices. Finally it passes through the four base points of orthogonal lines dropped from each Euler point to the face not containing the vertex that generated the Euler point.<ref>{{Cite book |last1=Outudee |first1=Somluck |last2=New |first2=Stephen |title=The Various Kinds of Centres of Simplices |publisher=Dept of Mathematics, Chulalongkorn University, Bangkok |url=http://www.math.sc.chula.ac.th/ICAA2002/pages/Somluck_Outudee.pdf |url-status=bot: unknown |archive-url=https://web.archive.org/web/20090227143222/http://www.math.sc.chula.ac.th/ICAA2002/pages/Somluck_Outudee.pdf |archive-date=27 February 2009}}</ref>

The center ''T'' of the twelve-point sphere also lies on the Euler line. Unlike its triangular counterpart, this center lies one third of the way from the Monge point ''M'' towards the circumcenter. Also, an orthogonal line through ''T'' to a chosen face is coplanar with two other orthogonal lines to the same face. The first is an orthogonal line passing through the corresponding Euler point to the chosen face. The second is an orthogonal line passing through the centroid of the chosen face. This orthogonal line through the twelve-point center lies midway between the Euler point orthogonal line and the centroidal orthogonal line. Furthermore, for any face, the twelve-point center lies at the midpoint of the corresponding Euler point and the orthocenter for that face.

The radius of the twelve-point sphere is one third of the circumradius of the reference tetrahedron.

There is a relation among the angles made by the faces of a general tetrahedron given by<ref>{{cite web |first=Daniel |last=Audet |title=Déterminants sphérique et hyperbolique de Cayley-Menger |url=http://archimede.mat.ulaval.ca/amq/bulletins/mai11/Chronique_note_math.mai11.pdf |publisher=Bulletin AMQ |date=May 2011 }}</ref>
:<math>\begin{vmatrix}  -1 & \cos{(\alpha_{12})} & \cos{(\alpha_{13})} & \cos{(\alpha_{14})}\\
\cos{(\alpha_{12})} & -1 & \cos{(\alpha_{23})} & \cos{(\alpha_{24})} \\
\cos{(\alpha_{13})} & \cos{(\alpha_{23})} & -1 & \cos{(\alpha_{34})} \\
\cos{(\alpha_{14})} & \cos{(\alpha_{24})} & \cos{(\alpha_{34})} & -1 \\ \end{vmatrix} = 0\,</math>

where ''α{{sub|ij}}'' is the angle between the faces ''i'' and ''j''.

The [[geometric median]] of the vertex position coordinates of a tetrahedron and its isogonic center are associated, under circumstances analogous to those observed for a triangle.  [[Lorenz Lindelöf]] found that, corresponding to any given tetrahedron is a point now known as an isogonic center, ''O'', at which the solid angles subtended by the faces are equal, having a common value of π [[Steradian|sr]], and at which the angles subtended by opposite edges are equal.<ref>{{cite journal |last1=Lindelof |first1=L. |title=Sur les maxima et minima d'une fonction des rayons vecteurs menés d'un point mobile à plusieurs centres fixes |journal=Acta Societatis Scientiarum Fennicae |date=1867 |volume=8 |issue=Part 1 |pages=189–203}}</ref>  A solid angle of π sr is one quarter of that subtended by all of space.  When all the solid angles at the vertices of a tetrahedron are smaller than π sr, ''O'' lies inside the tetrahedron, and because the sum of distances from ''O'' to the vertices is a minimum, ''O'' coincides with the [[geometric median]], ''M'', of the vertices.  In the event that the solid angle at one of the vertices, ''v'', measures exactly π sr, then ''O'' and ''M'' coincide with ''v''.  If however, a tetrahedron has a vertex, ''v'', with solid angle greater than π sr, ''M'' still corresponds to ''v'', but ''O'' lies outside the tetrahedron.