Editing Euler line (section)

==Generalizations==

===Quadrilateral===

In a [[Quadrilateral#Remarkable points and lines in a convex quadrilateral|convex quadrilateral]], the quasiorthocenter ''H'', the "area centroid" ''G'', and the [[quasicircumcenter]] ''O'' are [[collinear]] in this order on the Euler line, and ''HG'' = 2''GO''.<ref>{{citation
 | last = Myakishev | first = Alexei
 | journal = Forum Geometricorum
 | pages = 289–295
 | title = On Two Remarkable Lines Related to a Quadrilateral
 | url = http://forumgeom.fau.edu/FG2006volume6/FG200634.pdf
 | volume = 6
 | year = 2006}}.</ref>

===Tetrahedron===
{{Main article|Tetrahedron#Properties analogous to those of a triangle}}

A [[tetrahedron]] is a [[three-dimensional space|three-dimensional]] object bounded by four triangular [[face (geometry)|faces]].  Seven lines associated with a tetrahedron are concurrent at its centroid; its six midplanes intersect at its [[Monge point]]; and there is a circumsphere passing through all of the vertices, whose center is the circumcenter. These points define the "Euler line" of a tetrahedron analogous to that of a triangle. The centroid is the midpoint between its Monge point and circumcenter along this line. The center of the [[twelve-point sphere]] also lies on the Euler line.

===Simplicial polytope===

A [[simplicial polytope]] is a polytope whose facets are all [[simplex|simplices]] (plural of simplex). For example, every polygon is a simplicial polytope. The Euler line associated to such a polytope is the line determined by its centroid and [[circumcenter of mass]]. This definition of an Euler line generalizes the ones above.<ref>{{citation
 | last1 = Tabachnikov | first1 = Serge
 | last2 = Tsukerman | first2 = Emmanuel
 | date = May 2014
 | issue = 4
 | journal = [[Discrete and Computational Geometry]]
 | pages = 815–836
 | title = Circumcenter of Mass and Generalized Euler Line
 | doi=10.1007/s00454-014-9597-2
 | volume=51| arxiv = 1301.0496
 | s2cid = 12307207
 }}.</ref>

Suppose that <math>P</math> is a polygon. The Euler line <math>E</math> is sensitive to the symmetries of <math>P</math> in the following ways:

# If <math>P</math> has a line of reflection symmetry <math>L</math>, then <math>E</math> is either <math>L</math> or a point on <math>L</math>.
# If <math>P</math> has a center of rotational symmetry <math>C</math>, then <math>E=C</math>.