Euler line
In geometry, the Euler line, named after Leonhard Euler (/ˈɔɪlər/ OY-lər), 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.
Euler showed in 1765 that in any triangle, the orthocenter, circumcenter and centroid are collinear.
[1] However, the incenter generally does not lie on the Euler line;[3] it is on the Euler line only for isosceles triangles,[4] for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers.
447 [6]: p.104, #211, p.242, #346 The center of similitude of the orthic and tangential triangles is also on the Euler line.[5]: p.
satisfies the relation This follows from the fact that the absolute barycentric coordinates of
Further, the problem of Sylvester[7] reads as Now, using the vector addition, we deduce that By adding these three relations, term by term, we obtain that In conclusion,
In Dörrie's book,[7] the Euler line and the problem of Sylvester are put together into a single proof.
However, most of the proofs of the problem of Sylvester rely on the fundamental properties of free vectors, independently of the Euler line.
On the Euler line the centroid G is between the circumcenter O and the orthocenter H and is twice as far from the orthocenter as it is from the circumcenter:[6]: p.102 The segment GH is a diameter of the orthocentroidal circle.
The center N of the nine-point circle lies along the Euler line midway between the orthocenter and the circumcenter:[1] Thus the Euler line could be repositioned on a number line with the circumcenter O at the location 0, the centroid G at 2t, the nine-point center at 3t, and the orthocenter H at 6t for some scale factor t. Furthermore, the squared distance between the centroid and the circumcenter along the Euler line is less than the squared circumradius R2 by an amount equal to one-ninth the sum of the squares of the side lengths a, b, and c:[6]: p.71 In addition,[6]: p.102 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 An equation for the Euler line in barycentric coordinates
is[8] Another way to represent the Euler line is in terms of a parameter t. Starting with the circumcenter (with trilinear coordinates
every point on the Euler line, except the orthocenter, is given by the trilinear coordinates formed as a linear combination of the trilinears of these two points, for some t. For example: In a Cartesian coordinate system, denote the slopes of the sides of a triangle as
Then these slopes are related according to[9]: Lemma 1 Thus the slope of the Euler line (if finite) is expressible in terms of the slopes of the sides as Moreover, the Euler line is parallel to an acute triangle's side BC if and only if[9]: p.173
This is because the right triangle's orthocenter, the intersection of its altitudes, falls on the right-angled vertex while its circumcenter, the intersection of its perpendicular bisectors of sides, falls on the midpoint of the hypotenuse.
The Euler line of an isosceles triangle coincides with the axis of symmetry.
In an isosceles triangle the incenter falls on the Euler line.
[6]: p.111 In a convex quadrilateral, the quasiorthocenter H, the "area centroid" G, and the quasicircumcenter O are collinear in this order on the Euler line, and HG = 2GO.
[13] A tetrahedron is a three-dimensional object bounded by four triangular faces.
Seven lines associated with a tetrahedron are concurrent at its centroid; its six midplanes intersect at its Monge point; and there is a circumsphere passing through all of the vertices, whose center is the circumcenter.
These points define the "Euler line" of a tetrahedron analogous to that of a triangle.
The centroid is the midpoint between its Monge point and circumcenter along this line.
The center of the twelve-point sphere also lies on the Euler line.
in the following ways: A triangle's Kiepert parabola is the unique parabola that is tangent to the sides (two of them extended) of the triangle and has the Euler line as its directrix.[15]: p.
