Euler line

These points can be used to define an Euler line of a quadrilateral.

The line that passes through all of them is known as the Euler line.

However, the incenter lies on the Euler line only for isosceles triangles.

An interactive applet showing several triangle centers that lies on the Euler line.

The center of the tetrahedron's twelve-point sphere also lies on the Euler line.

If T represents the center of the twelve point sphere, then it also lies on the Euler line.

Any triangle and its Gossard triangle have the same Euler line.

Triangle centers on the Euler line, by Clark Kimberling.

The Euler line relationships are:

Let l be any line parallel to the Euler line of triangle ABC.

The de Longchamps point is the point of concurrence of several lines with the Euler line.

The orthocenter, along with the centroid, circumcenter and center of the nine-point circle all lie on a single line, known as the Euler line.

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.

A triangle's centroid lies on its Euler line between its orthocenter and its circumcenter, exactly twice as close to the latter as to the former.

The four Euler lines of an orthocentric system are orthogonal to the four orthic axes of an orthocentric system.

This fact can be used to prove that the Euler line of the intouch triangle of a triangle coincides with its OI line.

The Johnson triangle and its reference triangle share the same nine-point center, the same Euler line and the same nine-point circle.

The center of any nine-point circle (the nine-point center) lies on the corresponding triangle's Euler line, at the midpoint between that triangle's orthocenter and circumcenter.

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.

These points define the Euler line of the tetrahedron that is analogous to the Euler line of a triangle.

Note that the Euler line is orthogonal to the orthic axis and that the Soddy line is orthogonal to the Gergonne line.

Euler line of triangle ABC is the line passing through the centroid, the orthocenter and the circumcenter of triangle ABC.

Let ABC be the triangle formed by the Euler lines of the triangles BFD and CDE and similarly for the other two vertices.

"Non-Euclidean Triangle Continuum" by Robert A. Russell shows a non-Euclidean Euler line, the Wolfram Demonstrations Project, 2011.

As the reflection of the orthocenter around the circumcenter, the de Longchamps point belongs to the line through both of these points, which is the Euler line of the given triangle.