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Concept
Euler line
Summary
In geometry, the Euler line, named after Leonhard Euler (ˈɔɪlər), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.
The concept of a triangle's Euler line extends to the Euler line of other shapes, such as the quadrilateral and the tetrahedron.
Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear. This property is also true for another triangle center, 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 are all distinct from each other, and the Euler line is determined by any two of them.
Other notable points that lie on the Euler line include the de Longchamps point, the Schiffler point, the Exeter point, and the Gossard perspector. However, the incenter generally does not lie on the Euler line; it is on the Euler line only for isosceles triangles, for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers.
The tangential triangle of a reference triangle is tangent to the latter's circumcircle at the reference triangle's vertices. The circumcenter of the tangential triangle lies on the Euler line of the reference triangle. The center of similitude of the orthic and tangential triangles is also on the Euler line.
Let be a triangle. A proof of the fact that the circumcenter , the centroid and the orthocenter are collinear relies on free vectors. We start by stating the prerequisites. First, satisfies the relation
This follows from the fact that the absolute barycentric coordinates of are . Further, the problem of Sylvester reads as
Now, using the vector addition, we deduce that
By adding these three relations, term by term, we obtain that
In conclusion, , and so the three points , and (in this order) are collinear.
Official source
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