Editing Euler line (section)

===Individual centers===
Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are [[Line (geometry)|collinear]].<ref>{{cite journal
 | author = Euler, Leonhard
 | authorlink = Leonhard Euler
 | title = Solutio facilis problematum quorundam geometricorum difficillimorum
 |trans-title= Easy solution of some difficult geometric problems
 | journal = Novi Commentarii Academiae Scientarum Imperialis Petropolitanae
 | volume = 11
 | year = 1767
 | pages = 103–123
 | url = https://books.google.com/books?id=e1Y-AAAAcAAJ&pg=PA103 
 | id = <!--Enestrom number-->E325}} Reprinted in ''Opera Omnia'', ser. I, vol. XXVI, pp.&nbsp;139–157, Societas Scientiarum Naturalium Helveticae, Lausanne, 1953, {{MR|0061061}}.  Summarized at:  [http://math.dartmouth.edu/~euler/pages/E325.html Dartmouth College.]
</ref> This property is also true for another [[triangle center]], the [[nine-point center]], although it had not been defined in Euler's time. In equilateral triangles, these four points coincide, but in any other triangle they are all distinct from each other, and the Euler line is determined by any two of them.

Other notable points that lie on the Euler line include the [[de Longchamps point]], the [[Schiffler point]], the [[Exeter point]], and the [[Gossard perspector]].<ref name="k"/> However, the [[incenter]] generally does not lie on the Euler line;<ref>{{cite book | url=https://books.google.com/books?id=lR0SDnl2bPwC&pg=PA4 | title=Geometry Turned On: Dynamic Software in Learning, Teaching, and Research | publisher=The Mathematical Association of America |author1=Schattschneider, Doris |author2=King, James | year=1997 | pages=3–4 | isbn=978-0883850992}}</ref> it is on the Euler line only for [[isosceles triangle]]s,<ref>{{citation
 | last1 = Edmonds | first1 = Allan L.
 | last2 = Hajja | first2 = Mowaffaq
 | last3 = Martini | first3 = Horst
 | doi = 10.1007/s00025-008-0294-4
 | issue = 1–2
 | journal = [[Results in Mathematics]]
 | mr = 2430410
 | pages = 41–50
 | quote = It is well known that the incenter of a Euclidean triangle lies on its Euler line connecting the centroid and the circumcenter if and only if the triangle is isosceles
 | title = Orthocentric simplices and biregularity
 | volume = 52
 | year = 2008| s2cid = 121434528
 }}.</ref> for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers.

The [[tangential triangle]] of a reference triangle is tangent to the latter's [[circumcircle]] at the reference triangle's vertices. The circumcenter of the tangential triangle lies on the Euler line of the reference triangle.<ref name=SL/>{{rp|p. 447}} <ref name="ac"/>{{rp|p.104,#211;p.242,#346}} The [[center of similitude]] of the [[orthic triangle|orthic]] and tangential triangles is also on the Euler line.<ref name=SL>{{citation
 | last1 = Leversha | first1 = Gerry
 | last2 = Smith | first2 = G. C.
 | date = November 2007
 | issue = 522
 | journal = [[Mathematical Gazette]]
 | jstor = 40378417
 | pages = 436–452
 | title = Euler and triangle geometry
 | volume = 91| doi = 10.1017/S0025557200182087
 | s2cid = 125341434
 }}.</ref>{{rp|p. 447}}<ref name="ac"/>{{rp|p. 102}}