Euler line

Triangle

Under the Eulerian lines of a triangle (also Euler line, named after the mathematician Leonhard Euler ) is defined as the straight line passing through the centroid, the circumcenter and the orthocenter of the triangle. For the general tetrahedron in three-dimensional space there is the analogous term (see below).

It is also true with the focus S between the orthocenter H and circumcenter U lies. It also has 2 U H = 3 S. The resulting Euler straight line passes also through the center of the Feuerbach circle ( the same time the center of the track is ).

In an isosceles triangle, the Euler agrees precisely match the base belonging to the median line ( perpendicular bisector, altitude, angle bisector ). In the case of an equilateral triangle can no longer speak of the Euler's line, because then collapse the three decisive points S, U and H to a point. ( Otherwise you could indeed every line through one point to be construed as Euler's line, but what to avoid for the sake of clarity. )

On the Euler's line of triangle ABC is also the circumcenter of the triangle formed by the tangents to the circumcircle of triangle ABC at points A, B and C. In addition, contains the Euler Just another excellent points of the triangle, including the Longchamps Point and the Schiffler point.

Tetrahedron

For a general tetrahedron is called ( in analogy to the two-dimensional case of the triangle ), the Euler or Euler straight line ( engl. Euler line) of the straight line which connects the center of gravity and the center of the circumscribed sphere of.