# Euler's straight line

Euler straight line e (black),
height intersection H (red),
center of gravity S (green, intersection of the side bisectors),
circumcenter U (blue, intersection of the vertical lines),
Feuerbach circle with center N (black)

The Euler straight line or Euler straight line is a special straight line on the triangle , a so-called triangular transversal , on which a number of marked triangle points lie, including the center of gravity , the circumcenter , the height intersection and the center of the Feuerbach circle . It is named after the mathematician Leonhard Euler . For the general tetrahedron in three-dimensional space there is an analogous term (see below).

## properties

In a triangle, the center of gravity S , the vertical intersection point H and the circumcenter U lie on a common straight line, the Euler straight line. Since the center of the Feuerbach circle N is also the center of the line HU ( Feuerbach's theorem ), it is also on the Euler straight line. In addition, the following route relationships | apply to these four points HU | = 3 | US | = 6 | NS |, | HS | = 4 | NS |, | HN | = 3 | NS |, | NU | = 3 | NS | and | SU | = 2 | NS |.

Euler equation and Feuerbach equation

The following equations apply to the coordinates of the four points S, H, U and N:

• ${\ displaystyle 3 {\ vec {S}} = {\ vec {H}} + 2 {\ vec {U}}}$( Euler's equation )
• ${\ displaystyle 3 {\ vec {S}} = {\ vec {U}} + 2 {\ vec {N}}}$( Feuerbach equation )

In an isosceles triangle , Euler's straight line coincides with the side bisector belonging to the base ( vertical line , height , bisector ). In the case of an equilateral triangle one can no longer speak of the Euler straight line, because then the three determining points S , U and H coincide to one point. (Otherwise every straight line through this one point could be interpreted as Euler's straight line, but this is avoided for the sake of clarity.)

The circumcenter of the triangle, which is formed by the tangents to the circumference of triangle ABC at points A , B and C , also lies on the Euler straight line of triangle ABC . In addition, Euler's straight line contains other marked points of the triangle, including the Longchamps point , the Schiffler point and the Exeter point .

The center of the inscribed circle of the triangle lies on the Euler straight line when the triangle is isosceles.

## Tetrahedron

Euler straight line in black

For a general tetrahedron one calls (in analogy to the two-dimensional case of the triangle) the Euler straight line or Euler straight line from that straight line which connects the center of gravity of and the center of the sphere of . ${\ displaystyle {\ mathcal {T}} \ subset \ mathbb {R} ^ {3}}$${\ displaystyle {\ mathcal {T}}}$${\ displaystyle e ({\ mathcal {T}}) \ subset \ mathbb {R} ^ {3}}$ ${\ displaystyle S ({\ mathcal {T}})}$${\ displaystyle {\ mathcal {T}}}$ ${\ displaystyle U ({\ mathcal {T}})}$${\ displaystyle {\ mathcal {T}}}$

## Individual evidence

1. Allan L. Edmonds, Mowaffaq Hajja, Horst Martini: Ortho Centric simplices and Biregularity . In: Results in Mathematics . tape 52 , no. 1-2 , August 2008, ISSN  1422-6383 , p. 41–50 , doi : 10.1007 / s00025-008-0294-4 ( springer.com [accessed August 29, 2019]).
2. Altshiller-Court, p. 77