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. It passes through the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.

Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear. This property is also true for 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 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.

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

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

for some t.

Examples:

• centroid =
• nine-point center =
• De Longchamps point =
• Euler infinity point =