Euler line

Euler line

In geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral; it passes through several important points determined from the triangle. In the image, the Euler line is shown in red. It passes through the orthocenter (blue), the circumcenter (green), the centroid (orange), and the center of the nine-point circle (red) of the triangle.

Euler (1767) showed that in any triangle, the orthocenter, circumcenter, centroid, and nine-point center are collinear. In equilateral triangles, these four points coincide, but in any other triangle they do not, and the Euler line is determined by any two of them. The center of the nine-point circle lies midway along the Euler line between the orthocenter and the circumcenter, and the distance from the centroid to the circumcenter is half that from the centroid to the orthocenter.

Other notable points that lie on the Euler line are the de Longchamps point, the Schiffler point, the Exeter point and the far-out point. However, the incenter lies on the Euler line only for isosceles triangles.

The Euler line is its own complement, and therefore also its own anticomplement.

Let A, B, C denote the vertex angles of the reference triangle, and let x : y : z be a variable point in trilinear coordinates; then an equation for the Euler line is

:sin 2A sin(B - C)x + sin 2B sin(C - A)y + sin 2C sin(A - B)z = 0.

Another particularly useful way to represent the Euler line is in terms of a parameter t. Starting with the circumcenter (with trilinears cos A : cos B : cos C) and the orthocenter (with trilinears sec A : sec B : sec C = cos B cos C : cos C cos A : cos A cos B), every point on the Euler line, except the orthocenter, is given as

:cos A + t cos B cos C : cos B + t cos C cos A : cos C + t cos A cos B

for some t.

Examples:
* centroid = cos A + cos B cos C : cos B + cos C cos A : cos C + cos A cos B
* nine-point center = cos A + 2 cos B cos C : cos B + 2 cos C cos A : cos C + 2 cos A cos B
* De Longchamps point = cos A - cos B cos C : cos B - cos C cos A : cos C - cos A cos B
* Euler infinity point = cos A - 2 cos B cos C : cos B - 2 cos C cos A : cos C - 2 cos A cos B

References

*cite journal
author = Euler, Leonhard
authorlink = Leonhard Euler
title = Solutio facilis problematum quorundam geometricorum difficillimorum
journal = Novi Commentarii academiae scientarum imperialis Petropolitanae
volume = 11
year = 1767
pages = 103–123
url = http://math.dartmouth.edu/~euler/pages/E325.html
id = E325
Reprinted in "Opera Omnia", ser. I, vol. XXVI, pp. 139–157, Societas Scientiarum Naturalium Helveticae, Lausanne, 1953, MathSciNet | id = 0061061.

*cite journal
author = Kimberling, Clark
title = Triangle centers and central triangles
journal = Congressus Numerantium
volume = 129
year = 1998
pages = i–xxv, 1–295

External links

* [http://www.cut-the-knot.org/triangle/altEuler.shtml Altitudes and the Euler Line] and [http://www.cut-the-knot.org/triangle/EulerLine.shtml Euler Line and 9-Point Circle] at cut-the-knot
* [http://agutie.homestead.com/files/center/nine_point_center_euler.html Euler Line, Nine-Point Circle, and Nine-Point Center] Interactive illustration with 22 steps at Geometry from the Land of the Incas.
* [http://faculty.evansville.edu/ck6/tcenters/class/eulerline.html Triangle centers on the Euler line] , by Clark Kimberling.
* [http://www.uff.br/trianglecenters/euler-line.html An interactive Java applet showing several triangle centers that lies on the Euler line] .
*
* [http://demonstrations.wolfram.com/EulerLine/ "Euler Line"] by Eric Rowland, The Wolfram Demonstrations Project, 2007.


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Euler, Leonhard — born April 15, 1707, Basel, Switz. died Sept. 18, 1783, St. Petersburg, Russia Swiss mathematician. In 1733 he succeeded Daniel Bernoulli (see Bernoulli family) at the St. Petersburg Academy of Sciences. There he developed the theory of… …   Universalium

  • Euler'sche Gerade — Unter der eulerschen Geraden eines Dreiecks (auch Eulergeraden, benannt nach dem Mathematiker Leonhard Euler) versteht man die Gerade, die durch den Schwerpunkt, den Umkreismittelpunkt und den Höhenschnittpunkt des Dreiecks geht. Außerdem gilt ,… …   Deutsch Wikipedia

  • Euler–Mascheroni constant — Euler s constant redirects here. For the base of the natural logarithm, e ≈ 2.718..., see e (mathematical constant). The area of the blue region is equal to the Euler–Mascheroni constant. List of numbers – Irrational and suspected irrational… …   Wikipedia

  • Euler-Dreieck — Euler Zahlen als Koeffizienten von Euler Polynomen Die nach Leonhard Euler benannte Euler Zahl An,k in der Kombinatorik, auch geschrieben als E(n,k) oder , gibt die Anzahl der Permutationen (Anordnungen) von 1, …, n an, in denen genau k Elemente… …   Deutsch Wikipedia

  • Euler angles — The Euler angles were developed by Leonhard Euler to describe the orientation of a rigid body (a body in which the relative position of all its points is constant) in 3 dimensional Euclidean space. To give an object a specific orientation it may… …   Wikipedia

  • Euler class — In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how quot;twisted quot; the vector bundle… …   Wikipedia

  • Euler–Lagrange equation — In calculus of variations, the Euler–Lagrange equation, or Lagrange s equation is a differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and… …   Wikipedia

  • Euler method — In mathematics and computational science, the Euler method, named after Leonhard Euler, is a first order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic kind of explicit… …   Wikipedia

  • Euler-Konstante — γ Die Euler Mascheroni Konstante (nach den Mathematikern Leonhard Euler und Lorenzo Mascheroni), auch Eulersche Konstante, ist eine wichtige mathematische Konstante, die mit dem griechischen Buchstaben γ (gamma) bezeichnet wird. Ihre Definition… …   Deutsch Wikipedia

  • Euler spiral — A double end Euler spiral. An Euler spiral is a curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). Euler spirals are also commonly referred to as spiros,… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”