(
Mathematics - Geometry)
The
Euler line of a
triangle is defined to be a
line that passes through the following three points.
The three points coincide only for
equilateral triangles, in which case the line is not
well-defined. By the
theorem of Snapper, these points are always
collinear for any
triangle, and the
centroid is one-third the way from the
circumcenter to the
orthocenter. By the
nine point circle theorem, the nine-point circle center also lie on the
Euler line at the
midpoint of the
circumcenter and the
orthocenter.
There are 96 other centers of a triangle that also lie on the Euler line, out of the 1,114 centers listed at the encyclopedia of triangle centers at http://faculty.evansville.edu/ck6/encyclopedia/