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HandWiki. Euler Line. Encyclopedia. Available online: https://encyclopedia.pub/entry/30142 (accessed on 22 September 2026).
HandWiki. Euler Line. Encyclopedia. Available at: https://encyclopedia.pub/entry/30142. Accessed September 22, 2026.
HandWiki. "Euler Line" Encyclopedia, https://encyclopedia.pub/entry/30142 (accessed September 22, 2026).
HandWiki. (2022, October 19). Euler Line. In Encyclopedia. https://encyclopedia.pub/entry/30142
HandWiki. "Euler Line." Encyclopedia. Web. 19 October, 2022.
Euler Line
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The content is sourced from: https://handwiki.org/wiki/Euler_line

In geometry, the Euler line, named after Leonhard Euler (/ˈɔɪlər/), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle. The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron.

quadrilateral equilateral euler

References

  1. Euler, Leonhard (1767). "Solutio facilis problematum quorundam geometricorum difficillimorum". Novi Commentarii Academiae Scientarum Imperialis Petropolitanae 11: 103–123. E325. https://books.google.com/books?id=e1Y-AAAAcAAJ&pg=PA103#v=onepage&q&f=false.  Reprinted in Opera Omnia, ser. I, vol. XXVI, pp. 139–157, Societas Scientiarum Naturalium Helveticae, Lausanne, 1953, MR0061061. Summarized at: Dartmouth College.
  2. Kimberling, Clark (1998). "Triangle centers and central triangles". Congressus Numerantium 129: i–xxv, 1–295. 
  3. Schattschneider, Doris; King, James (1997). Geometry Turned On: Dynamic Software in Learning, Teaching, and Research. The Mathematical Association of America. pp. 3–4. ISBN 978-0883850992. https://books.google.com/books?id=lR0SDnl2bPwC&pg=PA4. 
  4. Edmonds, Allan L.; Hajja, Mowaffaq; Martini, Horst (2008), "Orthocentric simplices and biregularity", Results in Mathematics 52 (1–2): 41–50, doi:10.1007/s00025-008-0294-4, "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" . https://dx.doi.org/10.1007%2Fs00025-008-0294-4
  5. Leversha, Gerry; Smith, G. C. (November 2007), "Euler and triangle geometry", Mathematical Gazette 91 (522): 436–452 .
  6. Altshiller-Court, Nathan, College Geometry, Dover Publications, 2007 (orig. Barnes & Noble 1952).
  7. Dörrie, Heinrich, "100 Great Problems of Elementary Mathematics. Their History and Solution". Dover Publications, Inc., New York, 1965, ISBN:0-486-61348-8, pages 141 (Euler's Straight Line) and 142 (Problem of Sylvester)
  8. Scott, J.A., "Some examples of the use of areal coordinates in triangle geometry", Mathematical Gazette 83, November 1999, 472-477.
  9. Wladimir G. Boskoff, Laurent¸iu Homentcovschi, and Bogdan D. Suceava, "Gossard's Perspector and Projective Consequences", Forum Geometricorum, Volume 13 (2013), 169–184. [1]
  10. Francisco Javier Garc ́ıa Capita ́n, "Locus of Centroids of Similar Inscribed Triangles", Forum Geometricorum 16, 2016, 257–267 .http://forumgeom.fau.edu/FG2016volume16/FG201631.pdf
  11. Parry, C. F. (1991), "Steiner–Lehmus and the automedian triangle", The Mathematical Gazette 75 (472): 151–154 .
  12. Beluhov, Nikolai Ivanov. "Ten concurrent Euler lines", Forum Geometricorum 9, 2009, pp. 271–274. http://forumgeom.fau.edu/FG2009volume9/FG200924index.html
  13. Myakishev, Alexei (2006), "On Two Remarkable Lines Related to a Quadrilateral", Forum Geometricorum 6: 289–295, http://forumgeom.fau.edu/FG2006volume6/FG200634.pdf .
  14. Tabachnikov, Serge; Tsukerman, Emmanuel (May 2014), "Circumcenter of Mass and Generalized Euler Line", Discrete and Computational Geometry 51 (51): 815–836, doi:10.1007/s00454-014-9597-2 . https://dx.doi.org/10.1007%2Fs00454-014-9597-2
  15. Scimemi, Benedetto, "Simple Relations Regarding the Steiner Inellipse of a Triangle", Forum Geometricorum 10, 2010: 55–77. http://forumgeom.fau.edu/FG2010volume10/FG201008.pdf
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