Euler line

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The Euler line of a triangle (blue), and its relation to several triangle centers.

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 blue. It passes through the orthocenter (H), the circumcenter (O), the centroid (G), and the center of the nine-point circle (N) 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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \cos A : \cos B : \cos C ) and the orthocenter (with trilinears Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 = Failed to parse (Missing texvc executable; please see math/README to configure.): \cos A + \cos B \cos C : \cos B + \cos C \cos A : \cos C + \cos A \cos B

nine-point center = Failed to parse (Missing texvc executable; please see math/README to configure.): \cos A + 2 \cos B \cos C : \cos B + 2 \cos C \cos A : \cos C + 2 \cos A \cos B

De Longchamps point = Failed to parse (Missing texvc executable; please see math/README to configure.): \cos A - \cos B \cos C : \cos B - \cos C \cos A : \cos C - \cos A \cos B

Euler infinity point = Failed to parse (Missing texvc executable; please see math/README to configure.): \cos A - 2 \cos B \cos C : \cos B - 2 \cos C \cos A : \cos C - 2 \cos A \cos B

References

Euler, Leonhard (1767). "Solutio facilis problematum quorundam geometricorum difficillimorum". Novi Commentarii academiae scientarum imperialis Petropolitanae 11: 103–123. E325. Reprinted in Opera Omnia, ser. I, vol. XXVI, pp. 139–157, Societas Scientiarum Naturalium Helveticae, Lausanne, 1953, MR 0061061.