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.