Euler line (thing)

(Mathematics - Geometry)

The Euler line of a triangle is defined to be a line that passes through the following three points.

• the orthocenter (concurrent point of the three altitudes)
• the centroid (concurrent point of the three medians)
• the circumcenter (concurrent point of the three perpendicular bisectors)

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/